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bg-background","children":["$","$L39",null,{"backHref":"/a-level/maths/wjec/questions","basePath":"/a-level/maths/wjec/trigonometry-7/questions","formulaUrl":"https://www.wjec.co.uk/media/4jfdytcz/ss-w27-18-e.pdf","initialIndex":58,"initialQuestionWindow":[{"id":"83595e3c-e503-4372-8f45-0c7c60173098","revisionId":"b7eaedff-f322-49e5-b4d2-6c4383f2f8a6","questionNumber":"5","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":"7.1 Addition Formulae","calculatorUse":null,"isCalculation":true,"parts":[{"id":"864f4907-d937-4ebc-b2d3-92c49722f31e","label":"0","content":"Solve, for $0 \\leq x \\leq 360$, the equation,\n\n$$8 \\sin x \\cos x = 3$$\n\nGive your answers to 1 decimal place.","markscheme":"- Uses $2 \\sin x \\cos x = \\sin 2x$ to rewrite as $4 \\sin 2x = 3$ M1\n- $\\sin 2x = 0.75$ M1\n- Finds values for $2x$: $48.59, 131.41, 408.59, 491.41$ M1\n- Divides by 2 to find $x$ M1\n- Final answers: $24.3, 65.7, 204.3, 245.7$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Uses the double angle identity to obtain $4 \\sin 2x = 3$.","code":"M"},{"id":"p2","marks":1,"description":"Finds the principal value for $2x \\approx 48.6^\\circ$.","code":"M"},{"id":"p3","marks":1,"description":"Finds at least one other value for $2x$ in the range $[0, 720]$.","code":"M"},{"id":"p4","marks":1,"description":"Divides $2x$ values by 2 to find $x$.","code":"M"},{"id":"p5","marks":1,"description":"Provides all four correct answers: $24.3, 65.7, 204.3, 245.7$.","code":"A"}],"finalAnswers":["24.3","65.7","204.3","245.7"],"generatedAt":"2026-05-06T03:35:10.313Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"double_angle","label":"Double Angle Identity","steps":[{"type":"text","order":0,"title":"Use double angle identity","content":"The given equation is:\n\n$$8 \\sin x \\cos x = 3$$\n\nTo simplify the left-hand side, we recall the double-angle identity for sine:\n\n$$\\sin 2x = 2 \\sin x \\cos x$$\n\nWe can factor out a $4$ from the term on the left-hand side to match the identity:\n\n$$8 \\sin x \\cos x = 4(2 \\sin x \\cos x) = 4 \\sin 2x$$"},{"type":"text","order":1,"title":"Simplify the equation","content":"Substitute the identity back into the original equation:\n\n$$4 \\sin 2x = 3$$\n\nNow divide both sides by $4$ to isolate the trigonometric function:\n\n$$\\sin 2x = 0.75$$"},{"type":"text","order":2,"title":"Adjust the interval","content":"The original range for $x$ is given as:\n\n$$0^\\circ \\leq x \\leq 360^\\circ$$\n\nSince our trigonometric term has the argument $2x$, we must adjust the domain by multiplying all parts of the inequality by $2$:\n\n$$0^\\circ \\leq 2x \\leq 720^\\circ$$"},{"type":"text","order":3,"title":"Find primary solutions","content":"We solve $\\sin 2x = 0.75$ for $2x$ in the interval $[0^\\circ, 720^\\circ]$.\n\nFirst, find the principal value using the inverse sine function:\n\n$$2x = \\arcsin(0.75) \\approx 48.59^\\circ$$\n\nSince the sine function is positive in both the first and second quadrants, the second solution in the first cycle ($0^\\circ$ to $360^\\circ$) is:\n\n$$2x = 180^\\circ - 48.59^\\circ = 131.41^\\circ$$"},{"type":"text","order":4,"title":"Find remaining solutions","content":"Since our interval for $2x$ goes up to $720^\\circ$, we must find the corresponding solutions in the second full rotation. We do this by adding $360^\\circ$ to each of our first two solutions:\n\n$$2x = 48.59^\\circ + 360^\\circ = 408.59^\\circ$$\n\n$$2x = 131.41^\\circ + 360^\\circ = 491.41^\\circ$$\n\nThis gives us four values for $2x$ within our adjusted range:\n\n$$2x \\approx 48.59^\\circ, \\; 131.41^\\circ, \\; 408.59^\\circ, \\; 491.41^\\circ$$"},{"type":"text","order":5,"title":"Solve for x","content":"Finally, divide each value by $2$ to find $x$, and round to 1 decimal place:\n\n$$x_1 = \\frac{48.59^\\circ}{2} \\approx 24.3^\\circ$$\n\n$$x_2 = \\frac{131.41^\\circ}{2} \\approx 65.7^\\circ$$\n\n$$x_3 = \\frac{408.59^\\circ}{2} \\approx 204.3^\\circ$$\n\n$$x_4 = \\frac{491.41^\\circ}{2} \\approx 245.7^\\circ$$\n\nThus, the solutions are $x = 24.3^\\circ$, $65.7^\\circ$, $204.3^\\circ$, and $245.7^\\circ$.","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x), f2(x) = 3","instructions":"1. Set the calculator's angle setting to Degrees.\n2. Open the Graph tool and graph f1(x) = 8*sin(x)*cos(x) and f2(x) = 3.\n3. Set the window dimensions with x from 0 to 360.\n4. Use the Intersection tool to locate all intersection points."},"ti84":{"expression":"Y1 = 8*sin(X)*cos(X)\nY2 = 3","instructions":"1. Change MODE to DEGREE.\n2. Press [Y=] and enter Y1 = 8*sin(X)*cos(X) and Y2 = 3.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Ymin = -5, Ymax = 5.\n4. Press [GRAPH].\n5. Press [2nd] [CALC] and choose '5: intersect' to find the x-values of the intersections."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x);\\ y = 3","instructions":"1. Click the wrench icon, make sure 'Degrees' is selected.\n2. Set the x-axis range to [0, 360].\n3. Type: y = 8sin(x)cos(x)\n4. Type: y = 3\n5. Click on the intersection points to find their coordinate values."},"description":"Verify your solutions visually by graphing the original equation."}]}]},{"id":"graphical","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Set up the functions","content":"To solve the equation $8 \\sin x \\cos x = 3$ graphically, we can define two separate functions and find where their graphs intersect on the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n$$y_1 = 8 \\sin x \\cos x$$\n$$y_2 = 3$$"},{"type":"text","order":1,"title":"Configure the calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode since the domain is given in degrees ($0^\\circ \\leq x \\leq 360^\\circ$). Adjust your window settings to view the full domain:\n- **Xmin** $= 0$\n- **Xmax** $= 360$\n- **Ymin** $= -5$\n- **Ymax** $= 5$","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x)\nf2(x) = 3","instructions":"1. Go to the Graph mode.\n2. Set the angle unit to Degree in the settings.\n3. Input the functions:\n \$f1(x) = 8 \\sin(x) \\cos(x)\$\n \$f2(x) = 3\$\n4. Set the View Window: X from 0 to 360, Y from -5 to 5.\n5. Draw the graph."},"ti84":{"expression":"Y_1 = 8*sin(X)*cos(X)\nY_2 = 3","instructions":"1. Press [MODE] and ensure DEGREE is highlighted.\n2. Press [Y=] and enter:\n \$Y_1 = 8 \\sin(X) \\cos(X)\$\n \$Y_2 = 3\$\n3. Press [WINDOW] and set:\n Xmin = 0, Xmax = 360, Xscl = 30\n Ymin = -5, Ymax = 5, Yscl = 1\n4. Press [GRAPH] to display the functions."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x)\ny = 3","instructions":"1. Click on the wrench icon (Graph Settings) and select Degrees.\n2. Set the X-axis limits to [0, 360] and Y-axis limits to [-5, 5].\n3. Type: `y = 8*sin(x)*cos(x)`\n4. Type: `y = 3`"},"description":"Graph the equations and adjust the window settings."}]},{"type":"text","order":2,"title":"Find the intersections","content":"Use your graphing calculator's intersection tool to locate the coordinates of the points where the two curves meet. Within the range $[0, 360]$, you will see four intersection points.","calculator":[{"gdc":{"instructions":"1. Use the G-Solv (Graph Solve) tool.\n2. Select ISCT (Intersection).\n3. Use the arrow keys to cycle through and record all four intersection values."},"ti84":{"instructions":"1. Press [2nd] [CALC] and select option 5: intersect.\n2. Move the cursor near the first intersection point, then press [ENTER] three times. Write down the x-value.\n3. Repeat this process for the other three intersection points by moving the cursor close to each successive peak."},"desmos":{"instructions":"1. Click on the gray intersection points on the graph where the wave meets the line \$y = 3\$.\n2. Note down the x-coordinates displayed."},"description":"Find each of the 4 intersection points on the graph."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The calculator returns the following $x$-coordinates for the intersection points:\n\n* $x \\approx 24.295^\\circ$\n* $x \\approx 65.705^\\circ$\n* $x \\approx 204.295^\\circ$\n* $x \\approx 245.705^\\circ$\n\nRounding each answer to 1 decimal place as required, we obtain:\n\n$$x = 24.3^\\circ, \\; 65.7^\\circ, \\; 204.3^\\circ, \\; 245.7^\\circ$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]}],"generatedAt":"2026-06-09T07:08:37.695Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"864f4907-d937-4ebc-b2d3-92c49722f31e:double_angle","label":"Double Angle Identity","steps":[{"type":"text","order":0,"title":"Use double angle identity","content":"The given equation is:\n\n$$8 \\sin x \\cos x = 3$$\n\nTo simplify the left-hand side, we recall the double-angle identity for sine:\n\n$$\\sin 2x = 2 \\sin x \\cos x$$\n\nWe can factor out a $4$ from the term on the left-hand side to match the identity:\n\n$$8 \\sin x \\cos x = 4(2 \\sin x \\cos x) = 4 \\sin 2x$$"},{"type":"text","order":1,"title":"Simplify the equation","content":"Substitute the identity back into the original equation:\n\n$$4 \\sin 2x = 3$$\n\nNow divide both sides by $4$ to isolate the trigonometric function:\n\n$$\\sin 2x = 0.75$$"},{"type":"text","order":2,"title":"Adjust the interval","content":"The original range for $x$ is given as:\n\n$$0^\\circ \\leq x \\leq 360^\\circ$$\n\nSince our trigonometric term has the argument $2x$, we must adjust the domain by multiplying all parts of the inequality by $2$:\n\n$$0^\\circ \\leq 2x \\leq 720^\\circ$$"},{"type":"text","order":3,"title":"Find primary solutions","content":"We solve $\\sin 2x = 0.75$ for $2x$ in the interval $[0^\\circ, 720^\\circ]$.\n\nFirst, find the principal value using the inverse sine function:\n\n$$2x = \\arcsin(0.75) \\approx 48.59^\\circ$$\n\nSince the sine function is positive in both the first and second quadrants, the second solution in the first cycle ($0^\\circ$ to $360^\\circ$) is:\n\n$$2x = 180^\\circ - 48.59^\\circ = 131.41^\\circ$$"},{"type":"text","order":4,"title":"Find remaining solutions","content":"Since our interval for $2x$ goes up to $720^\\circ$, we must find the corresponding solutions in the second full rotation. We do this by adding $360^\\circ$ to each of our first two solutions:\n\n$$2x = 48.59^\\circ + 360^\\circ = 408.59^\\circ$$\n\n$$2x = 131.41^\\circ + 360^\\circ = 491.41^\\circ$$\n\nThis gives us four values for $2x$ within our adjusted range:\n\n$$2x \\approx 48.59^\\circ, \\; 131.41^\\circ, \\; 408.59^\\circ, \\; 491.41^\\circ$$"},{"type":"text","order":5,"title":"Solve for x","content":"Finally, divide each value by $2$ to find $x$, and round to 1 decimal place:\n\n$$x_1 = \\frac{48.59^\\circ}{2} \\approx 24.3^\\circ$$\n\n$$x_2 = \\frac{131.41^\\circ}{2} \\approx 65.7^\\circ$$\n\n$$x_3 = \\frac{408.59^\\circ}{2} \\approx 204.3^\\circ$$\n\n$$x_4 = \\frac{491.41^\\circ}{2} \\approx 245.7^\\circ$$\n\nThus, the solutions are $x = 24.3^\\circ$, $65.7^\\circ$, $204.3^\\circ$, and $245.7^\\circ$.","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x), f2(x) = 3","instructions":"1. Set the calculator's angle setting to Degrees.\n2. Open the Graph tool and graph f1(x) = 8*sin(x)*cos(x) and f2(x) = 3.\n3. Set the window dimensions with x from 0 to 360.\n4. Use the Intersection tool to locate all intersection points."},"ti84":{"expression":"Y1 = 8*sin(X)*cos(X)\nY2 = 3","instructions":"1. Change MODE to DEGREE.\n2. Press [Y=] and enter Y1 = 8*sin(X)*cos(X) and Y2 = 3.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Ymin = -5, Ymax = 5.\n4. Press [GRAPH].\n5. Press [2nd] [CALC] and choose '5: intersect' to find the x-values of the intersections."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x);\\ y = 3","instructions":"1. Click the wrench icon, make sure 'Degrees' is selected.\n2. Set the x-axis range to [0, 360].\n3. Type: y = 8sin(x)cos(x)\n4. Type: y = 3\n5. Click on the intersection points to find their coordinate values."},"description":"Verify your solutions visually by graphing the original equation."}]}]},{"id":"864f4907-d937-4ebc-b2d3-92c49722f31e:graphical","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Set up the functions","content":"To solve the equation $8 \\sin x \\cos x = 3$ graphically, we can define two separate functions and find where their graphs intersect on the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n$$y_1 = 8 \\sin x \\cos x$$\n$$y_2 = 3$$"},{"type":"text","order":1,"title":"Configure the calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode since the domain is given in degrees ($0^\\circ \\leq x \\leq 360^\\circ$). Adjust your window settings to view the full domain:\n- **Xmin** $= 0$\n- **Xmax** $= 360$\n- **Ymin** $= -5$\n- **Ymax** $= 5$","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x)\nf2(x) = 3","instructions":"1. Go to the Graph mode.\n2. Set the angle unit to Degree in the settings.\n3. Input the functions:\n \$f1(x) = 8 \\sin(x) \\cos(x)\$\n \$f2(x) = 3\$\n4. Set the View Window: X from 0 to 360, Y from -5 to 5.\n5. Draw the graph."},"ti84":{"expression":"Y_1 = 8*sin(X)*cos(X)\nY_2 = 3","instructions":"1. Press [MODE] and ensure DEGREE is highlighted.\n2. Press [Y=] and enter:\n \$Y_1 = 8 \\sin(X) \\cos(X)\$\n \$Y_2 = 3\$\n3. Press [WINDOW] and set:\n Xmin = 0, Xmax = 360, Xscl = 30\n Ymin = -5, Ymax = 5, Yscl = 1\n4. Press [GRAPH] to display the functions."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x)\ny = 3","instructions":"1. Click on the wrench icon (Graph Settings) and select Degrees.\n2. Set the X-axis limits to [0, 360] and Y-axis limits to [-5, 5].\n3. Type: `y = 8*sin(x)*cos(x)`\n4. Type: `y = 3`"},"description":"Graph the equations and adjust the window settings."}]},{"type":"text","order":2,"title":"Find the intersections","content":"Use your graphing calculator's intersection tool to locate the coordinates of the points where the two curves meet. Within the range $[0, 360]$, you will see four intersection points.","calculator":[{"gdc":{"instructions":"1. Use the G-Solv (Graph Solve) tool.\n2. Select ISCT (Intersection).\n3. Use the arrow keys to cycle through and record all four intersection values."},"ti84":{"instructions":"1. Press [2nd] [CALC] and select option 5: intersect.\n2. Move the cursor near the first intersection point, then press [ENTER] three times. Write down the x-value.\n3. Repeat this process for the other three intersection points by moving the cursor close to each successive peak."},"desmos":{"instructions":"1. Click on the gray intersection points on the graph where the wave meets the line \$y = 3\$.\n2. Note down the x-coordinates displayed."},"description":"Find each of the 4 intersection points on the graph."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The calculator returns the following $x$-coordinates for the intersection points:\n\n* $x \\approx 24.295^\\circ$\n* $x \\approx 65.705^\\circ$\n* $x \\approx 204.295^\\circ$\n* $x \\approx 245.705^\\circ$\n\nRounding each answer to 1 decimal place as required, we obtain:\n\n$$x = 24.3^\\circ, \\; 65.7^\\circ, \\; 204.3^\\circ, \\; 245.7^\\circ$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]}],"generatedAt":"2026-06-09T07:08:50.985Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"a2eec4d2-89c8-415c-8c8b-6b1e523138f7","revisionId":"7b5db209-7a88-4deb-a4d4-79d74de733dd","questionNumber":"6","questionType":"LA","specification":"Prove that:","difficulty":1,"difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":"6.4 Trigonometric Identities","calculatorUse":null,"isCalculation":true,"parts":[{"id":"efc6aa51-8568-4ccd-81e5-b4a1d048ba68","label":"a","content":"$$\\sec^4x - \\tan^4x \\equiv 1 + 2\\tan^2x$","markscheme":"- Factorises as difference of two squares: $(\\sec^2 x + \\tan^2 x)(\\sec^2 x - \\tan^2 x)$ M1\n- Uses $\\sec^2 x - \\tan^2 x = 1$ and $\\sec^2 x = 1 + \\tan^2 x$ M1\n- Simplifies to $(1 + \\tan^2 x + \\tan^2 x)(1) = 1 + 2 \\tan^2 x$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Factorises the LHS using difference of two squares.","code":"M"},{"id":"p2","marks":1,"description":"Applies the identity $\\sec^2x = 1 + \\tan^2x$ to reach the RHS.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T04:14:47.574Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"difference_of_squares","label":"Difference of Squares","steps":[{"type":"text","order":0,"title":"Identify the expression","content":"We want to prove the identity:\n\n$$\\sec^4x - \\tan^4x \\equiv 1 + 2\\tan^2x$$\n\nTo do this, we start by looking at the Left-Hand Side (LHS):\n\n$$\\text{LHS} = \\sec^4x - \\tan^4x$$\n\nNotice that both terms are perfect squares: $\\sec^4x = (\\sec^2x)^2$ and $\\tan^4x = (\\tan^2x)^2$. This means we can factorise the expression using the **difference of two squares** algebraic identity:\n\n$$A^2 - B^2 = (A + B)(A - B)$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=lhs}{\\sec^4x - \\tan^4x} \\equiv (\\htmlData{anchor=sec2}{\\sec^2x})^2 - (\\htmlData{anchor=tan2}{\\tan^2x})^2$"},{"latex":"$$\\phantom{\\sec^4x - \\tan^4x} \\equiv (\\htmlData{anchor=sec-fact}{\\sec^2x} + \\htmlData{anchor=tan-fact}{\\tan^2x})(\\sec^2x - \\tan^2x)$"}],"highlights":[{"color":"sky","style":"underline","anchor":"sec2"},{"color":"amber","style":"underline","anchor":"tan2"}],"connections":[{"to":"sec-fact","from":"sec2","color":"sky","label":null,"style":"solid"},{"to":"tan-fact","from":"tan2","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Factorise the LHS","description":"Factorising the difference of two squares earns the first method mark (M1)."},{"type":"text","order":2,"title":"Recall key trigonometric identities","content":"To simplify our factorised expression, we use the fundamental Pythagorean identity relating secant and tangent:\n\n$$\\sec^2x \\equiv 1 + \\tan^2x$$\n\nBy subtracting $\\tan^2x$ from both sides, we also get:\n\n$$\\sec^2x - \\tan^2x \\equiv 1$$\n\nThese two substitutions are crucial for simplifying the terms in our brackets (earning the second mark **M1**)."},{"type":"text","order":3,"title":"Substitute and simplify","content":"Now, we substitute these identities back into our factorised LHS:\n\n$$\\begin{aligned} (\\sec^2x + \\tan^2x)(\\sec^2x - \\tan^2x) &= (1 + \\tan^2x + \\tan^2x)(1) \\\\ &= 1 + 2\\tan^2x \\end{aligned}$$\n\nThis is exactly equal to the Right-Hand Side (RHS), completing our proof and earning the final accuracy mark (**A1**):\n\n$$\\sec^4x - \\tan^4x \\equiv 1 + 2\\tan^2x$$"},{"type":"text","order":4,"title":"Substitute identity into Part 2","content":"Now we move on to **Part 2**. We need to solve the equation:\n\n$$\\sec^4x - \\tan^4x = 3 \\quad \\text{for} \\quad 0^\\circ \\leq x \\leq 360^\\circ$$\n\nUsing the identity we proved in Part 1, we can substitute the left-hand side with $1 + 2\\tan^2x$ (earning the first method mark **M1**):\n\n$$1 + 2\\tan^2x = 3$$\n\nSubtract $1$ from both sides:\n\n$$2\\tan^2x = 2$$\n\nDivide by $2$:\n\n$$\\tan^2x = 1$$\n\nTaking the square root of both sides gives two cases (earning **M1**):\n\n$$\\tan x = 1 \\quad \\text{or} \\quad \\tan x = -1$$"},{"type":"text","order":5,"title":"Solve for x","content":"We solve for $x$ in the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n**Case 1:** $\\tan x = 1$\n- The principal value is $x = \\tan^{-1}(1) = 45^\\circ$.\n- Since tangent has a period of $180^\\circ$, the other solution in the range is:\n $$x = 45^\\circ + 180^\\circ = 225^\\circ$$\nThis earns the accuracy mark **A1**.\n\n**Case 2:** $\\tan x = -1$\n- The principal value is $x = \\tan^{-1}(-1) = -45^\\circ$ (which is outside our range).\n- Adding $180^\\circ$ to find the positive solutions inside the range:\n $$x = -45^\\circ + 180^\\circ = 135^\\circ$$\n $$x = 135^\\circ + 180^\\circ = 315^\\circ$$\nThis earns the second accuracy mark **A1**.\n\nSo, the final solutions are:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]},{"id":"direct_substitution","label":"Direct Substitution and Expansion","steps":[{"type":"text","order":0,"title":"Identify the identity","content":"To prove the identity, we will start with the Left-Hand Side (LHS) and transform it to look like the Right-Hand Side (RHS):\n\n$$\\text{LHS} = \\sec^4x - \\tan^4x$$\n\nWe can recall the fundamental trigonometric identity relating $\\sec^2 x$ and $\\tan^2 x$:\n\n$$\\sec^2 x = 1 + \\tan^2 x$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\htmlData{anchor=sec2}{\\color{@sky}{\\sec^2 x}})^2 - \\tan^4 x$"},{"latex":"$$(\\htmlData{anchor=subst}{\\color{@sky}{1 + \\tan^2 x}})^2 - \\tan^4 x$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sec2"},{"color":"sky","style":"text-color","anchor":"subst"}],"connections":[{"to":"subst","from":"sec2","color":"sky","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute the identity","description":"We write $\\sec^4x$ as $(\\sec^2x)^2$ and substitute $\\sec^2 x = 1 + \\tan^2 x$ directly into the expression."},{"type":"text","order":2,"title":"Expand and simplify","content":"Now we expand the squared bracket $(1 + \\tan^2 x)^2$:\n\n$$(1 + \\tan^2 x)^2 = 1 + 2\\tan^2 x + \\tan^4 x$$\n\nSubstituting this back into our expression:\n\n$$\\begin{aligned} \\text{LHS} &= (1 + 2\\tan^2 x + \\tan^4 x) - \\tan^4 x \\\\ &= 1 + 2\\tan^2 x \\end{aligned}$$\n\nThis simplified expression is exactly the Right-Hand Side (RHS), completing the proof."},{"type":"text","order":3,"title":"Substitute identity into Part 2","content":"Now let's solve **Part 2** of the question:\n\n$$\\sec^4x - \\tan^4x = 3 \\quad \\text{for} \\quad 0^\\circ \\leq x \\leq 360^\\circ$$\n\nWe substitute the identity $1 + 2\\tan^2x$ from Part 1 into the equation:\n\n$$1 + 2\\tan^2x = 3$$\n\nSubtracting $1$ and dividing by $2$ gives:\n\n$$\\tan^2x = 1$$\n\nThus:\n\n$$\\tan x = \\pm 1$$"},{"type":"text","order":4,"title":"Find all solutions for x","content":"We find the solutions for $\\tan x = \\pm 1$ in the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n- For $\\tan x = 1$:\n $$x = 45^\\circ, \\quad x = 180^\\circ + 45^\\circ = 225^\\circ$$\n\n- For $\\tan x = -1$:\n $$x = 180^\\circ - 45^\\circ = 135^\\circ, \\quad x = 360^\\circ - 45^\\circ = 315^\\circ$$\n\nTherefore, the complete set of solutions is:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]},{"id":"sine_cosine_conversion","label":"Conversion to Sine and Cosine","steps":[{"type":"text","order":0,"title":"Convert LHS to sine and cosine","content":"To prove the identity using the conversion to sine and cosine, we begin by writing the left-hand side (LHS) in terms of $\\sin x$ and $\\cos x$.\n\nRecall the definitions:\n$$\\sec x = \\frac{1}{\\cos x} \\quad \\text{and} \\quad \\tan x = \\frac{\\sin x}{\\cos x}$$\n\nSubstituting these into the LHS:\n$$\\sec^4x - \\tan^4x = \\left(\\frac{1}{\\cos x}\\right)^4 - \\left(\\frac{\\sin x}{\\cos x}\\right)^4 = \\frac{1}{\\cos^4 x} - \\frac{\\sin^4 x}{\\cos^4 x}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{\\htmlData{anchor=num}{1 - \\sin^4 x}}{\\cos^4 x}$"},{"latex":"$$\\frac{\\htmlData{anchor=fact}{(1 - \\sin^2 x)(1 + \\sin^2 x)}}{\\cos^4 x}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"num"},{"color":"sky","style":"text-color","anchor":"fact"}],"connections":[{"to":"fact","from":"num","color":"sky","label":"factorise","style":"solid"}]},"order":1,"title":"Combine and factorise","description":"Combine the fractions over their common denominator and use the difference of two squares identity, $a^2 - b^2 = (a - b)(a + b)$, to factorise the numerator."},{"type":"text","order":2,"title":"Simplify using Pythagorean identity","content":"Using the Pythagorean identity $\\cos^2 x = 1 - \\sin^2 x$, we can substitute this into our numerator:\n\n$$\\frac{(1 - \\sin^2 x)(1 + \\sin^2 x)}{\\cos^4 x} = \\frac{\\cos^2 x (1 + \\sin^2 x)}{\\cos^4 x}$$\n\nNow, cancel $\\cos^2 x$ from both the numerator and denominator:\n\n$$\\frac{1 + \\sin^2 x}{\\cos^2 x}$$"},{"type":"text","order":3,"title":"Convert RHS to sine and cosine","content":"Now, let's write the right-hand side (RHS) in terms of $\\sin x$ and $\\cos x$ to show that both sides are identical:\n\n$$\\text{RHS} = 1 + 2\\tan^2 x = 1 + 2\\left(\\frac{\\sin^2 x}{\\cos^2 x}\\right)$$\n\nWrite this as a single fraction over the common denominator $\\cos^2 x$:\n\n$$\\text{RHS} = \\frac{\\cos^2 x + 2\\sin^2 x}{\\cos^2 x}$$"},{"type":"text","order":4,"title":"Simplify RHS and conclude","content":"Substitute the Pythagorean identity $\\cos^2 x = 1 - \\sin^2 x$ into the RHS numerator:\n\n$$\\text{RHS} = \\frac{(1 - \\sin^2 x) + 2\\sin^2 x}{\\cos^2 x} = \\frac{1 + \\sin^2 x}{\\cos^2 x}$$\n\nSince both the LHS and RHS simplify to the exact same expression:\n\n$$\\text{LHS} \\equiv \\text{RHS} \\equiv \\frac{1 + \\sin^2 x}{\\cos^2 x}$$\n\nThe identity is proved."},{"type":"text","order":5,"title":"Substitute identity into Part 2","content":"Using the identity $1 + 2\\tan^2 x$ we proved in Part 1, we can solve the equation for **Part 2**:\n\n$$\\sec^4x - \\tan^4x = 3 \\quad \\text{for} \\quad 0^\\circ \\leq x \\leq 360^\\circ$$\n\nSubstituting the identity gives:\n\n$$1 + 2\\tan^2x = 3$$\n\nThis simplifies directly to:\n\n$$\\tan^2x = 1 \\implies \\tan x = \\pm 1$$"},{"type":"text","order":6,"title":"Find the angles","content":"In the range $0^\\circ \\leq x \\leq 360^\\circ$, we solve for $\\tan x = \\pm 1$:\n\n- When $\\tan x = 1$:\n $$x = 45^\\circ \\quad \\text{and} \\quad x = 225^\\circ$$\n- When $\\tan x = -1$:\n $$x = 135^\\circ \\quad \\text{and} \\quad x = 315^\\circ$$\n\nThe full solution set is:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]}],"generatedAt":"2026-06-09T07:50:42.124Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"b6035d5c-1f84-461e-b004-d9c87da4bbac","label":"b","content":"Hence solve, for $0 \\leq x \\leq 360$, the equation,\n\n$$\\sec^4x - \\tan^4x = 3$$","markscheme":"- Uses result from (a): $1 + 2 \\tan^2 x = 3$ M1\n- Simplifies to $\\tan^2 x = 1 \\Rightarrow \\tan x = \\pm 1$ M1\n- Finds $x = 45^\\circ, 225^\\circ$ (for $\\tan x = 1$) A1\n- Finds $x = 135^\\circ, 315^\\circ$ (for $\\tan x = -1$) A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Equates the identity result to 3.","code":"M"},{"id":"p2","marks":1,"description":"Solves to find $\\tan x = \\pm 1$.","code":"M"},{"id":"p3","marks":1,"description":"Finds at least two correct solutions.","code":"A"},{"id":"p4","marks":1,"description":"Finds all four correct solutions: $45, 135, 225, 315$.","code":"A"}],"finalAnswers":["45","135","225","315"],"generatedAt":"2026-05-06T04:14:47.574Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"using_identity","label":"Using Part (a) Identity","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$ \\sec^4 x - \\tan^4 x \\equiv 1 + 2\\tan^2 x $$"},{"latex":"$$$ \\htmlData{anchor=eq-lhs}{\\color{@sky}{\\sec^4 x - \\tan^4 x}} = 3 $$"},{"latex":"$$$ \\htmlData{anchor=eq-sub}{\\color{@sky}{1 + 2\\tan^2 x}} = 3 $$"}],"highlights":[{"color":"sky","style":"box","anchor":"eq-lhs"},{"color":"sky","style":"box","anchor":"eq-sub"}],"connections":[{"to":"eq-sub","from":"eq-lhs","color":"sky","label":"Substitute","style":"solid"}]},"order":0,"title":"Apply the identity","description":"We are given the equation $\\sec^4 x - \\tan^4 x = 3$. In part (a), we proved the identity $\\sec^4 x - \\tan^4 x \\equiv 1 + 2\\tan^2 x$. We can substitute this identity directly into our equation."},{"type":"text","order":1,"title":"Solve for tan x","content":"Now, we solve the simplified equation for $\\tan x$:\n\n$$\\begin{aligned} 1 + 2\\tan^2 x &= 3 \\\\ 2\\tan^2 x &= 2 \\\\ \\tan^2 x &= 1 \\\\ \\tan x &= \\pm 1 \\end{aligned}$$\n\nThis gives us two cases to solve: $\\tan x = 1$ and $\\tan x = -1$."},{"type":"text","order":2,"title":"Find solutions for tan x = 1","content":"For the first case, $\\tan x = 1$:\n\n- The principal value is $x = \\arctan(1) = 45^\\circ$.\n- Since the tangent function has a period of $180^\\circ$, we find the second solution within the interval $0^\\circ \\le x \\le 360^\\circ$ by adding $180^\\circ$:\n $$x = 45^\\circ + 180^\\circ = 225^\\circ$$"},{"type":"text","order":3,"title":"Find solutions for tan x = -1","content":"For the second case, $\\tan x = -1$:\n\n- The primary positive value in our interval is $x = 180^\\circ - 45^\\circ = 135^\\circ$.\n- Adding $180^\\circ$ to find the other solution in the interval gives:\n $$x = 135^\\circ + 180^\\circ = 315^\\circ$$"},{"type":"text","order":4,"title":"State final solutions","content":"Combining the solutions from both cases, the final values of $x$ in the interval $0^\\circ \\le x \\le 360^\\circ$ are:\n\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$","calculator":[{"gdc":{"instructions":"1. Set angle units to degrees.\n2. Graph the functions \$f(x) = (\\cos(x))^{-4} - (\\tan(x))^4\$ and \$g(x) = 3\$.\n3. Use the intersection tool to find the coordinate intersections within the domain \$[0, 360]\$."},"ti84":{"instructions":"1. Ensure your calculator is in DEGREE mode.\n2. Press [Y=] and enter \$Y_1 = (1/\\cos(X))^4 - (\\tan(X))^4\$ and \$Y_2 = 3\$.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Ymin = -1, Ymax = 5.\n4. Press [GRAPH] to view the curves.\n5. Press [2nd] [TRACE] and select '5: intersect' to find the intersection points at X = 45, 135, 225, and 315."},"desmos":{"instructions":"1. Set the grid to Degree mode under graph settings.\n2. Type `y = (sec(x))^4 - (tan(x))^4` in the first row.\n3. Type `y = 3` in the second row.\n4. Set the x-axis limits from 0 to 360.\n5. Click on the gray intersection points to reveal the solutions: \$x = 45^\\circ\$, \$135^\\circ\$, \$225^\\circ\$, and \$315^\\circ\$."},"description":"Verify the solutions by finding the intersections of $y = \\sec^4 x - \\tan^4 x$ and $y = 3$ on a graphing calculator."}]}]},{"id":"factorization","label":"Factorization as Difference of Squares","steps":[{"type":"text","order":0,"title":"Factorize using difference of squares","content":"To solve the equation \n\n$$\\sec^4x - \\tan^4x = 3$$\n\nwe can recognize the left-hand side as a difference of two squares, since $\\sec^4x = (\\sec^2x)^2$ and $\\tan^4x = (\\tan^2x)^2$. \n\nRecall the algebraic difference of squares formula:\n\n$$a^2 - b^2 = (a - b)(a + b)$$\n\nApplying this identity where $a = \\sec^2x$ and $b = \\tan^2x$, we can factorize the equation as:\n\n$$(\\sec^2 x - \\tan^2 x)(\\sec^2 x + \\tan^2 x) = 3$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\htmlData{anchor=pythag-factor}{\\color{@sky}{\\sec^2 x - \\tan^2 x}})(\\sec^2 x + \\tan^2 x) = 3$"},{"text":"Using the fundamental Pythagorean identity:"},{"latex":"$$1 + \\tan^2 x = \\sec^2 x \\implies \\htmlData{anchor=pythag-value}{\\color{@sky}{1}} = \\sec^2 x - \\tan^2 x$"},{"text":"Substituting this back into our factored expression gives:"},{"latex":"$$\\htmlData{anchor=sub-one}{\\color{@sky}{1}} \\cdot (\\sec^2 x + \\tan^2 x) = 3$"},{"latex":"$$\\sec^2 x + \\tan^2 x = 3$"}],"highlights":[{"color":"sky","style":"box","anchor":"pythag-factor"},{"color":"sky","style":"text-color","anchor":"pythag-value"}],"connections":[{"to":"pythag-factor","from":"pythag-value","color":"sky","label":"Pythagorean Identity","style":"solid"},{"to":"sub-one","from":"pythag-value","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Apply Pythagorean identity","description":"We simplify the first factor using the key trigonometric identity connecting secant and tangent."},{"type":"text","order":2,"title":"Rewrite in terms of tangent","content":"Now we have a simpler equation: \n\n$$\\sec^2 x + \\tan^2 x = 3$$\n\nTo solve for $x$, we should write everything in terms of a single trigonometric function. We can substitute the identity $\\sec^2 x = 1 + \\tan^2 x$ into the equation:\n\n$$\\begin{aligned} (1 + \\tan^2 x) + \\tan^2 x &= 3 \\\\ 1 + 2\\tan^2 x &= 3 \\end{aligned}$$\n\nThis matches the form shown in the markscheme to earn the first method mark M1."},{"type":"text","order":3,"title":"Solve for tangent","content":"Now, we isolate $\\tan x$:\n\n$$\\begin{aligned} 2\\tan^2 x &= 2 \\\\ \\tan^2 x &= 1 \\\\ \\tan x &= \\pm 1 \\end{aligned}$$\n\nThis simplification to $\\tan x = \\pm 1$ earns the second method mark M1."},{"type":"text","order":4,"title":"Find the solutions for x","content":"We need to find all solutions in the interval $0^\\circ \\le x \\le 360^\\circ$.\n\n**Case 1: $\\tan x = 1$**\n- The principal value is $x = \\arctan(1) = 45^\\circ$.\n- Tangent has a period of $180^\\circ$, so the second solution in our interval is $x = 45^\\circ + 180^\\circ = 225^\\circ$.\n\nThis earns the third mark A1.\n\n**Case 2: $\\tan x = -1$**\n- The solutions are found in the second and fourth quadrants:\n- $x = 180^\\circ - 45^\\circ = 135^\\circ$\n- $x = 360^\\circ - 45^\\circ = 315^\\circ$\n\nThis earns the final accuracy mark A1.\n\nThus, the complete set of solutions is:\n\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]},{"id":"sin_cos_conversion","label":"Conversion to Sine and Cosine","steps":[{"type":"text","order":0,"title":"Rewrite using sine and cosine","content":"To solve the equation using the **Conversion to Sine and Cosine** approach, we first express both $\\sec x$ and $\\tan x$ in terms of $\\sin x$ and $\\cos x$.\n\nRecall the fundamental trigonometric definitions:\n$$\\sec x = \\frac{1}{\\cos x} \\quad \\text{and} \\quad \\tan x = \\frac{\\sin x}{\\cos x}$$\n\nSubstituting these into our equation, we get:\n$$\\left(\\frac{1}{\\cos x}\\right)^4 - \\left(\\frac{\\sin x}{\\cos x}\\right)^4 = 3$$\n$$\\frac{1}{\\cos^4 x} - \\frac{\\sin^4 x}{\\cos^4 x} = 3$$"},{"type":"text","order":1,"title":"Combine and simplify the fraction","content":"Since the fractions have a common denominator, we can combine them into a single fraction:\n$$\\frac{1 - \\sin^4 x}{\\cos^4 x} = 3$$\n\nNext, we simplify the numerator using the difference of two squares. Notice that $1 - \\sin^4 x$ can be factored as:\n$$1 - \\sin^4 x = (1^2)^2 - (\\sin^2 x)^2 = (1 - \\sin^2 x)(1 + \\sin^2 x)$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$1 - \\sin^2 x = \\htmlData{anchor=cos-identity}{\\color{@sky}{\\cos^2 x}}$"},{"latex":"$$1 - \\sin^4 x = \\htmlData{anchor=cos-sub}{\\color{@sky}{\\cos^2 x}}(1 + \\sin^2 x)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"cos-identity"},{"color":"sky","style":"text-color","anchor":"cos-sub"}],"connections":[{"to":"cos-sub","from":"cos-identity","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Substitute the Pythagorean identity","description":"Using the Pythagorean identity $1 - \\sin^2 x = \\cos^2 x$, we can substitute this into our factored expression for the numerator."},{"type":"text","order":3,"title":"Simplify the equation","content":"Now, substitute the simplified expression for the numerator back into the main equation:\n$$\\frac{\\cos^2 x(1 + \\sin^2 x)}{\\cos^4 x} = 3$$\n\nWe can cancel $\\cos^2 x$ from both the numerator and denominator (noting that $\\cos x \\neq 0$ for $\\sec x$ and $\\tan x$ to be defined):\n$$\\frac{1 + \\sin^2 x}{\\cos^2 x} = 3$$\n\nMultiply both sides by $\\cos^2 x$:\n$$1 + \\sin^2 x = 3\\cos^2 x$$"},{"type":"text","order":4,"title":"Solve for sine","content":"To solve this equation, let's express everything in terms of $\\sin^2 x$ by replacing $\\cos^2 x$ with $1 - \\sin^2 x$:\n$$\\begin{aligned} 1 + \\sin^2 x &= 3(1 - \\sin^2 x) \\\\ 1 + \\sin^2 x &= 3 - 3\\sin^2 x \\\\ 4\\sin^2 x &= 2 \\\\ \\sin^2 x &= \\frac{1}{2} \\end{aligned}$$\n\nTaking the square root of both sides gives:\n$$\\sin x = \\pm\\sqrt{\\frac{1}{2}} = \\pm\\frac{1}{\\sqrt{2}}$$"},{"type":"text","order":5,"title":"Find the final angles","content":"We need to find all solutions in the interval $0^\\circ \\leq x \\leq 360^\\circ$.\n\nFor $\\sin x = \\frac{1}{\\sqrt{2}}$ (positive in Quadrants I and II):\n- $x = 45^\\circ$\n- $x = 180^\\circ - 45^\\circ = 135^\\circ$\n\nFor $\\sin x = -\\frac{1}{\\sqrt{2}}$ (negative in Quadrants III and IV):\n- $x = 180^\\circ + 45^\\circ = 225^\\circ$\n- $x = 360^\\circ - 45^\\circ = 315^\\circ$\n\nThus, the four solutions are:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]}],"generatedAt":"2026-06-11T03:50:11.680Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"efc6aa51-8568-4ccd-81e5-b4a1d048ba68:difference_of_squares","label":"(a) Difference of Squares","steps":[{"type":"text","order":0,"title":"Identify the expression","content":"We want to prove the identity:\n\n$$\\sec^4x - \\tan^4x \\equiv 1 + 2\\tan^2x$$\n\nTo do this, we start by looking at the Left-Hand Side (LHS):\n\n$$\\text{LHS} = \\sec^4x - \\tan^4x$$\n\nNotice that both terms are perfect squares: $\\sec^4x = (\\sec^2x)^2$ and $\\tan^4x = (\\tan^2x)^2$. This means we can factorise the expression using the **difference of two squares** algebraic identity:\n\n$$A^2 - B^2 = (A + B)(A - B)$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=lhs}{\\sec^4x - \\tan^4x} \\equiv (\\htmlData{anchor=sec2}{\\sec^2x})^2 - (\\htmlData{anchor=tan2}{\\tan^2x})^2$"},{"latex":"$$\\phantom{\\sec^4x - \\tan^4x} \\equiv (\\htmlData{anchor=sec-fact}{\\sec^2x} + \\htmlData{anchor=tan-fact}{\\tan^2x})(\\sec^2x - \\tan^2x)$"}],"highlights":[{"color":"sky","style":"underline","anchor":"sec2"},{"color":"amber","style":"underline","anchor":"tan2"}],"connections":[{"to":"sec-fact","from":"sec2","color":"sky","label":null,"style":"solid"},{"to":"tan-fact","from":"tan2","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Factorise the LHS","description":"Factorising the difference of two squares earns the first method mark (M1)."},{"type":"text","order":2,"title":"Recall key trigonometric identities","content":"To simplify our factorised expression, we use the fundamental Pythagorean identity relating secant and tangent:\n\n$$\\sec^2x \\equiv 1 + \\tan^2x$$\n\nBy subtracting $\\tan^2x$ from both sides, we also get:\n\n$$\\sec^2x - \\tan^2x \\equiv 1$$\n\nThese two substitutions are crucial for simplifying the terms in our brackets (earning the second mark **M1**)."},{"type":"text","order":3,"title":"Substitute and simplify","content":"Now, we substitute these identities back into our factorised LHS:\n\n$$\\begin{aligned} (\\sec^2x + \\tan^2x)(\\sec^2x - \\tan^2x) &= (1 + \\tan^2x + \\tan^2x)(1) \\\\ &= 1 + 2\\tan^2x \\end{aligned}$$\n\nThis is exactly equal to the Right-Hand Side (RHS), completing our proof and earning the final accuracy mark (**A1**):\n\n$$\\sec^4x - \\tan^4x \\equiv 1 + 2\\tan^2x$$"},{"type":"text","order":4,"title":"Substitute identity into Part 2","content":"Now we move on to **Part 2**. We need to solve the equation:\n\n$$\\sec^4x - \\tan^4x = 3 \\quad \\text{for} \\quad 0^\\circ \\leq x \\leq 360^\\circ$$\n\nUsing the identity we proved in Part 1, we can substitute the left-hand side with $1 + 2\\tan^2x$ (earning the first method mark **M1**):\n\n$$1 + 2\\tan^2x = 3$$\n\nSubtract $1$ from both sides:\n\n$$2\\tan^2x = 2$$\n\nDivide by $2$:\n\n$$\\tan^2x = 1$$\n\nTaking the square root of both sides gives two cases (earning **M1**):\n\n$$\\tan x = 1 \\quad \\text{or} \\quad \\tan x = -1$$"},{"type":"text","order":5,"title":"Solve for x","content":"We solve for $x$ in the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n**Case 1:** $\\tan x = 1$\n- The principal value is $x = \\tan^{-1}(1) = 45^\\circ$.\n- Since tangent has a period of $180^\\circ$, the other solution in the range is:\n $$x = 45^\\circ + 180^\\circ = 225^\\circ$$\nThis earns the accuracy mark **A1**.\n\n**Case 2:** $\\tan x = -1$\n- The principal value is $x = \\tan^{-1}(-1) = -45^\\circ$ (which is outside our range).\n- Adding $180^\\circ$ to find the positive solutions inside the range:\n $$x = -45^\\circ + 180^\\circ = 135^\\circ$$\n $$x = 135^\\circ + 180^\\circ = 315^\\circ$$\nThis earns the second accuracy mark **A1**.\n\nSo, the final solutions are:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]},{"id":"efc6aa51-8568-4ccd-81e5-b4a1d048ba68:direct_substitution","label":"(a) Direct Substitution and Expansion","steps":[{"type":"text","order":0,"title":"Identify the identity","content":"To prove the identity, we will start with the Left-Hand Side (LHS) and transform it to look like the Right-Hand Side (RHS):\n\n$$\\text{LHS} = \\sec^4x - \\tan^4x$$\n\nWe can recall the fundamental trigonometric identity relating $\\sec^2 x$ and $\\tan^2 x$:\n\n$$\\sec^2 x = 1 + \\tan^2 x$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\htmlData{anchor=sec2}{\\color{@sky}{\\sec^2 x}})^2 - \\tan^4 x$"},{"latex":"$$(\\htmlData{anchor=subst}{\\color{@sky}{1 + \\tan^2 x}})^2 - \\tan^4 x$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sec2"},{"color":"sky","style":"text-color","anchor":"subst"}],"connections":[{"to":"subst","from":"sec2","color":"sky","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute the identity","description":"We write $\\sec^4x$ as $(\\sec^2x)^2$ and substitute $\\sec^2 x = 1 + \\tan^2 x$ directly into the expression."},{"type":"text","order":2,"title":"Expand and simplify","content":"Now we expand the squared bracket $(1 + \\tan^2 x)^2$:\n\n$$(1 + \\tan^2 x)^2 = 1 + 2\\tan^2 x + \\tan^4 x$$\n\nSubstituting this back into our expression:\n\n$$\\begin{aligned} \\text{LHS} &= (1 + 2\\tan^2 x + \\tan^4 x) - \\tan^4 x \\\\ &= 1 + 2\\tan^2 x \\end{aligned}$$\n\nThis simplified expression is exactly the Right-Hand Side (RHS), completing the proof."},{"type":"text","order":3,"title":"Substitute identity into Part 2","content":"Now let's solve **Part 2** of the question:\n\n$$\\sec^4x - \\tan^4x = 3 \\quad \\text{for} \\quad 0^\\circ \\leq x \\leq 360^\\circ$$\n\nWe substitute the identity $1 + 2\\tan^2x$ from Part 1 into the equation:\n\n$$1 + 2\\tan^2x = 3$$\n\nSubtracting $1$ and dividing by $2$ gives:\n\n$$\\tan^2x = 1$$\n\nThus:\n\n$$\\tan x = \\pm 1$$"},{"type":"text","order":4,"title":"Find all solutions for x","content":"We find the solutions for $\\tan x = \\pm 1$ in the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n- For $\\tan x = 1$:\n $$x = 45^\\circ, \\quad x = 180^\\circ + 45^\\circ = 225^\\circ$$\n\n- For $\\tan x = -1$:\n $$x = 180^\\circ - 45^\\circ = 135^\\circ, \\quad x = 360^\\circ - 45^\\circ = 315^\\circ$$\n\nTherefore, the complete set of solutions is:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]},{"id":"efc6aa51-8568-4ccd-81e5-b4a1d048ba68:sine_cosine_conversion","label":"(a) Conversion to Sine and Cosine","steps":[{"type":"text","order":0,"title":"Convert LHS to sine and cosine","content":"To prove the identity using the conversion to sine and cosine, we begin by writing the left-hand side (LHS) in terms of $\\sin x$ and $\\cos x$.\n\nRecall the definitions:\n$$\\sec x = \\frac{1}{\\cos x} \\quad \\text{and} \\quad \\tan x = \\frac{\\sin x}{\\cos x}$$\n\nSubstituting these into the LHS:\n$$\\sec^4x - \\tan^4x = \\left(\\frac{1}{\\cos x}\\right)^4 - \\left(\\frac{\\sin x}{\\cos x}\\right)^4 = \\frac{1}{\\cos^4 x} - \\frac{\\sin^4 x}{\\cos^4 x}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{\\htmlData{anchor=num}{1 - \\sin^4 x}}{\\cos^4 x}$"},{"latex":"$$\\frac{\\htmlData{anchor=fact}{(1 - \\sin^2 x)(1 + \\sin^2 x)}}{\\cos^4 x}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"num"},{"color":"sky","style":"text-color","anchor":"fact"}],"connections":[{"to":"fact","from":"num","color":"sky","label":"factorise","style":"solid"}]},"order":1,"title":"Combine and factorise","description":"Combine the fractions over their common denominator and use the difference of two squares identity, $a^2 - b^2 = (a - b)(a + b)$, to factorise the numerator."},{"type":"text","order":2,"title":"Simplify using Pythagorean identity","content":"Using the Pythagorean identity $\\cos^2 x = 1 - \\sin^2 x$, we can substitute this into our numerator:\n\n$$\\frac{(1 - \\sin^2 x)(1 + \\sin^2 x)}{\\cos^4 x} = \\frac{\\cos^2 x (1 + \\sin^2 x)}{\\cos^4 x}$$\n\nNow, cancel $\\cos^2 x$ from both the numerator and denominator:\n\n$$\\frac{1 + \\sin^2 x}{\\cos^2 x}$$"},{"type":"text","order":3,"title":"Convert RHS to sine and cosine","content":"Now, let's write the right-hand side (RHS) in terms of $\\sin x$ and $\\cos x$ to show that both sides are identical:\n\n$$\\text{RHS} = 1 + 2\\tan^2 x = 1 + 2\\left(\\frac{\\sin^2 x}{\\cos^2 x}\\right)$$\n\nWrite this as a single fraction over the common denominator $\\cos^2 x$:\n\n$$\\text{RHS} = \\frac{\\cos^2 x + 2\\sin^2 x}{\\cos^2 x}$$"},{"type":"text","order":4,"title":"Simplify RHS and conclude","content":"Substitute the Pythagorean identity $\\cos^2 x = 1 - \\sin^2 x$ into the RHS numerator:\n\n$$\\text{RHS} = \\frac{(1 - \\sin^2 x) + 2\\sin^2 x}{\\cos^2 x} = \\frac{1 + \\sin^2 x}{\\cos^2 x}$$\n\nSince both the LHS and RHS simplify to the exact same expression:\n\n$$\\text{LHS} \\equiv \\text{RHS} \\equiv \\frac{1 + \\sin^2 x}{\\cos^2 x}$$\n\nThe identity is proved."},{"type":"text","order":5,"title":"Substitute identity into Part 2","content":"Using the identity $1 + 2\\tan^2 x$ we proved in Part 1, we can solve the equation for **Part 2**:\n\n$$\\sec^4x - \\tan^4x = 3 \\quad \\text{for} \\quad 0^\\circ \\leq x \\leq 360^\\circ$$\n\nSubstituting the identity gives:\n\n$$1 + 2\\tan^2x = 3$$\n\nThis simplifies directly to:\n\n$$\\tan^2x = 1 \\implies \\tan x = \\pm 1$$"},{"type":"text","order":6,"title":"Find the angles","content":"In the range $0^\\circ \\leq x \\leq 360^\\circ$, we solve for $\\tan x = \\pm 1$:\n\n- When $\\tan x = 1$:\n $$x = 45^\\circ \\quad \\text{and} \\quad x = 225^\\circ$$\n- When $\\tan x = -1$:\n $$x = 135^\\circ \\quad \\text{and} \\quad x = 315^\\circ$$\n\nThe full solution set is:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]},{"id":"b6035d5c-1f84-461e-b004-d9c87da4bbac:using_identity","label":"(b) Using Part (a) Identity","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$ \\sec^4 x - \\tan^4 x \\equiv 1 + 2\\tan^2 x $$"},{"latex":"$$$ \\htmlData{anchor=eq-lhs}{\\color{@sky}{\\sec^4 x - \\tan^4 x}} = 3 $$"},{"latex":"$$$ \\htmlData{anchor=eq-sub}{\\color{@sky}{1 + 2\\tan^2 x}} = 3 $$"}],"highlights":[{"color":"sky","style":"box","anchor":"eq-lhs"},{"color":"sky","style":"box","anchor":"eq-sub"}],"connections":[{"to":"eq-sub","from":"eq-lhs","color":"sky","label":"Substitute","style":"solid"}]},"order":0,"title":"Apply the identity","description":"We are given the equation $\\sec^4 x - \\tan^4 x = 3$. In part (a), we proved the identity $\\sec^4 x - \\tan^4 x \\equiv 1 + 2\\tan^2 x$. We can substitute this identity directly into our equation."},{"type":"text","order":1,"title":"Solve for tan x","content":"Now, we solve the simplified equation for $\\tan x$:\n\n$$\\begin{aligned} 1 + 2\\tan^2 x &= 3 \\\\ 2\\tan^2 x &= 2 \\\\ \\tan^2 x &= 1 \\\\ \\tan x &= \\pm 1 \\end{aligned}$$\n\nThis gives us two cases to solve: $\\tan x = 1$ and $\\tan x = -1$."},{"type":"text","order":2,"title":"Find solutions for tan x = 1","content":"For the first case, $\\tan x = 1$:\n\n- The principal value is $x = \\arctan(1) = 45^\\circ$.\n- Since the tangent function has a period of $180^\\circ$, we find the second solution within the interval $0^\\circ \\le x \\le 360^\\circ$ by adding $180^\\circ$:\n $$x = 45^\\circ + 180^\\circ = 225^\\circ$$"},{"type":"text","order":3,"title":"Find solutions for tan x = -1","content":"For the second case, $\\tan x = -1$:\n\n- The primary positive value in our interval is $x = 180^\\circ - 45^\\circ = 135^\\circ$.\n- Adding $180^\\circ$ to find the other solution in the interval gives:\n $$x = 135^\\circ + 180^\\circ = 315^\\circ$$"},{"type":"text","order":4,"title":"State final solutions","content":"Combining the solutions from both cases, the final values of $x$ in the interval $0^\\circ \\le x \\le 360^\\circ$ are:\n\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$","calculator":[{"gdc":{"instructions":"1. Set angle units to degrees.\n2. Graph the functions \$f(x) = (\\cos(x))^{-4} - (\\tan(x))^4\$ and \$g(x) = 3\$.\n3. Use the intersection tool to find the coordinate intersections within the domain \$[0, 360]\$."},"ti84":{"instructions":"1. Ensure your calculator is in DEGREE mode.\n2. Press [Y=] and enter \$Y_1 = (1/\\cos(X))^4 - (\\tan(X))^4\$ and \$Y_2 = 3\$.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Ymin = -1, Ymax = 5.\n4. Press [GRAPH] to view the curves.\n5. Press [2nd] [TRACE] and select '5: intersect' to find the intersection points at X = 45, 135, 225, and 315."},"desmos":{"instructions":"1. Set the grid to Degree mode under graph settings.\n2. Type `y = (sec(x))^4 - (tan(x))^4` in the first row.\n3. Type `y = 3` in the second row.\n4. Set the x-axis limits from 0 to 360.\n5. Click on the gray intersection points to reveal the solutions: \$x = 45^\\circ\$, \$135^\\circ\$, \$225^\\circ\$, and \$315^\\circ\$."},"description":"Verify the solutions by finding the intersections of $y = \\sec^4 x - \\tan^4 x$ and $y = 3$ on a graphing calculator."}]}]},{"id":"b6035d5c-1f84-461e-b004-d9c87da4bbac:factorization","label":"(b) Factorization as Difference of Squares","steps":[{"type":"text","order":0,"title":"Factorize using difference of squares","content":"To solve the equation \n\n$$\\sec^4x - \\tan^4x = 3$$\n\nwe can recognize the left-hand side as a difference of two squares, since $\\sec^4x = (\\sec^2x)^2$ and $\\tan^4x = (\\tan^2x)^2$. \n\nRecall the algebraic difference of squares formula:\n\n$$a^2 - b^2 = (a - b)(a + b)$$\n\nApplying this identity where $a = \\sec^2x$ and $b = \\tan^2x$, we can factorize the equation as:\n\n$$(\\sec^2 x - \\tan^2 x)(\\sec^2 x + \\tan^2 x) = 3$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\htmlData{anchor=pythag-factor}{\\color{@sky}{\\sec^2 x - \\tan^2 x}})(\\sec^2 x + \\tan^2 x) = 3$"},{"text":"Using the fundamental Pythagorean identity:"},{"latex":"$$1 + \\tan^2 x = \\sec^2 x \\implies \\htmlData{anchor=pythag-value}{\\color{@sky}{1}} = \\sec^2 x - \\tan^2 x$"},{"text":"Substituting this back into our factored expression gives:"},{"latex":"$$\\htmlData{anchor=sub-one}{\\color{@sky}{1}} \\cdot (\\sec^2 x + \\tan^2 x) = 3$"},{"latex":"$$\\sec^2 x + \\tan^2 x = 3$"}],"highlights":[{"color":"sky","style":"box","anchor":"pythag-factor"},{"color":"sky","style":"text-color","anchor":"pythag-value"}],"connections":[{"to":"pythag-factor","from":"pythag-value","color":"sky","label":"Pythagorean Identity","style":"solid"},{"to":"sub-one","from":"pythag-value","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Apply Pythagorean identity","description":"We simplify the first factor using the key trigonometric identity connecting secant and tangent."},{"type":"text","order":2,"title":"Rewrite in terms of tangent","content":"Now we have a simpler equation: \n\n$$\\sec^2 x + \\tan^2 x = 3$$\n\nTo solve for $x$, we should write everything in terms of a single trigonometric function. We can substitute the identity $\\sec^2 x = 1 + \\tan^2 x$ into the equation:\n\n$$\\begin{aligned} (1 + \\tan^2 x) + \\tan^2 x &= 3 \\\\ 1 + 2\\tan^2 x &= 3 \\end{aligned}$$\n\nThis matches the form shown in the markscheme to earn the first method mark M1."},{"type":"text","order":3,"title":"Solve for tangent","content":"Now, we isolate $\\tan x$:\n\n$$\\begin{aligned} 2\\tan^2 x &= 2 \\\\ \\tan^2 x &= 1 \\\\ \\tan x &= \\pm 1 \\end{aligned}$$\n\nThis simplification to $\\tan x = \\pm 1$ earns the second method mark M1."},{"type":"text","order":4,"title":"Find the solutions for x","content":"We need to find all solutions in the interval $0^\\circ \\le x \\le 360^\\circ$.\n\n**Case 1: $\\tan x = 1$**\n- The principal value is $x = \\arctan(1) = 45^\\circ$.\n- Tangent has a period of $180^\\circ$, so the second solution in our interval is $x = 45^\\circ + 180^\\circ = 225^\\circ$.\n\nThis earns the third mark A1.\n\n**Case 2: $\\tan x = -1$**\n- The solutions are found in the second and fourth quadrants:\n- $x = 180^\\circ - 45^\\circ = 135^\\circ$\n- $x = 360^\\circ - 45^\\circ = 315^\\circ$\n\nThis earns the final accuracy mark A1.\n\nThus, the complete set of solutions is:\n\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]},{"id":"b6035d5c-1f84-461e-b004-d9c87da4bbac:sin_cos_conversion","label":"(b) Conversion to Sine and Cosine","steps":[{"type":"text","order":0,"title":"Rewrite using sine and cosine","content":"To solve the equation using the **Conversion to Sine and Cosine** approach, we first express both $\\sec x$ and $\\tan x$ in terms of $\\sin x$ and $\\cos x$.\n\nRecall the fundamental trigonometric definitions:\n$$\\sec x = \\frac{1}{\\cos x} \\quad \\text{and} \\quad \\tan x = \\frac{\\sin x}{\\cos x}$$\n\nSubstituting these into our equation, we get:\n$$\\left(\\frac{1}{\\cos x}\\right)^4 - \\left(\\frac{\\sin x}{\\cos x}\\right)^4 = 3$$\n$$\\frac{1}{\\cos^4 x} - \\frac{\\sin^4 x}{\\cos^4 x} = 3$$"},{"type":"text","order":1,"title":"Combine and simplify the fraction","content":"Since the fractions have a common denominator, we can combine them into a single fraction:\n$$\\frac{1 - \\sin^4 x}{\\cos^4 x} = 3$$\n\nNext, we simplify the numerator using the difference of two squares. Notice that $1 - \\sin^4 x$ can be factored as:\n$$1 - \\sin^4 x = (1^2)^2 - (\\sin^2 x)^2 = (1 - \\sin^2 x)(1 + \\sin^2 x)$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$1 - \\sin^2 x = \\htmlData{anchor=cos-identity}{\\color{@sky}{\\cos^2 x}}$"},{"latex":"$$1 - \\sin^4 x = \\htmlData{anchor=cos-sub}{\\color{@sky}{\\cos^2 x}}(1 + \\sin^2 x)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"cos-identity"},{"color":"sky","style":"text-color","anchor":"cos-sub"}],"connections":[{"to":"cos-sub","from":"cos-identity","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Substitute the Pythagorean identity","description":"Using the Pythagorean identity $1 - \\sin^2 x = \\cos^2 x$, we can substitute this into our factored expression for the numerator."},{"type":"text","order":3,"title":"Simplify the equation","content":"Now, substitute the simplified expression for the numerator back into the main equation:\n$$\\frac{\\cos^2 x(1 + \\sin^2 x)}{\\cos^4 x} = 3$$\n\nWe can cancel $\\cos^2 x$ from both the numerator and denominator (noting that $\\cos x \\neq 0$ for $\\sec x$ and $\\tan x$ to be defined):\n$$\\frac{1 + \\sin^2 x}{\\cos^2 x} = 3$$\n\nMultiply both sides by $\\cos^2 x$:\n$$1 + \\sin^2 x = 3\\cos^2 x$$"},{"type":"text","order":4,"title":"Solve for sine","content":"To solve this equation, let's express everything in terms of $\\sin^2 x$ by replacing $\\cos^2 x$ with $1 - \\sin^2 x$:\n$$\\begin{aligned} 1 + \\sin^2 x &= 3(1 - \\sin^2 x) \\\\ 1 + \\sin^2 x &= 3 - 3\\sin^2 x \\\\ 4\\sin^2 x &= 2 \\\\ \\sin^2 x &= \\frac{1}{2} \\end{aligned}$$\n\nTaking the square root of both sides gives:\n$$\\sin x = \\pm\\sqrt{\\frac{1}{2}} = \\pm\\frac{1}{\\sqrt{2}}$$"},{"type":"text","order":5,"title":"Find the final angles","content":"We need to find all solutions in the interval $0^\\circ \\leq x \\leq 360^\\circ$.\n\nFor $\\sin x = \\frac{1}{\\sqrt{2}}$ (positive in Quadrants I and II):\n- $x = 45^\\circ$\n- $x = 180^\\circ - 45^\\circ = 135^\\circ$\n\nFor $\\sin x = -\\frac{1}{\\sqrt{2}}$ (negative in Quadrants III and IV):\n- $x = 180^\\circ + 45^\\circ = 225^\\circ$\n- $x = 360^\\circ - 45^\\circ = 315^\\circ$\n\nThus, the four solutions are:\n$$x = 45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ$$"}]}],"generatedAt":"2026-06-11T03:50:11.747Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"01d13dfe-e432-4a4b-9945-ad584dc111e6","revisionId":"9aa0e10c-6351-4ecc-ad10-25ed984dd6e2","questionNumber":null,"questionType":"LA","specification":"","difficulty":null,"difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"a24d7ae1-f759-4091-a8dd-4a2bf8f382db","label":"a","content":"Use the identity $\\cos^2\\theta + \\sin^2\\theta \\equiv 1$ to prove that $\\tan^2\\theta \\equiv \\sec^2\\theta - 1$.","markscheme":"- Divides $\\cos^2\\theta + \\sin^2\\theta \\equiv 1$ throughout by $\\cos^2\\theta$: $1 + \\tan^2\\theta \\equiv \\sec^2\\theta$ M1\n- $\\tan^2\\theta \\equiv \\sec^2\\theta - 1$ as required A1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"c70c35fa-6b1c-418c-9c51-c5926d193cba","label":"b","content":"Solve, for $0° \\leq \\theta \\leq 360°$, the equation\n\n$\\tan^2\\theta + \\sec^2\\theta + 5\\sec\\theta = 2$\n\nGive your answers to one decimal place.","markscheme":"- Substitutes to give $\\left(\\sec^2\\theta - 1\\right) + \\sec^2\\theta + 5\\sec\\theta = 2$ M1\n- $2\\sec^2\\theta + 5\\sec\\theta - 3 = 0$ A1\n- $(2\\sec\\theta - 1)(\\sec\\theta + 3) = 0$ M1\n- $\\sec\\theta = \\frac{1}{2}$ gives $\\cos\\theta = 2$, which is impossible, so this root is rejected A1\n- $\\sec\\theta = -3$ gives $\\cos\\theta = -\\frac{1}{3}$, so $\\theta = 109.5°$ and $\\theta = 250.5°$ A1\n\nThe rejection must be seen for the fourth 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Trigonometry (A-level only)","topicId":"30ac4c05-8d09-4ec4-ac0a-b9371d6b1f6e","topicSidebar":"$L3a","topicSlug":"trigonometry-7","topicTitle":"3.5 Trigonometry (A-level only)"}]}],"$L3b","$L3c"]}]

3.5 Trigonometry (A-level only)

Question 55

3.5 Trigonometry (A-level only) Questions

3.5 Trigonometry (A-level only)

Question 55

3.5 Trigonometry (A-level only) Questions