494894","title":"1.5.16 Proof involving trigonometric functions (A-level only) Flashcards","topicId":"e6b17fd8-580a-42cc-9869-28c177fa2161","topicSlug":"proof-involving-trigonometric-functions","cards":[{"id":"cb079e7b-b4fa-4a70-9865-86e345e9c5df","cardType":"basic","front":"Why is $\\sin x=\\cos x$ an equation rather than an identity?","back":"It is true only for particular values of $x$.","points":1,"mcqOptions":null},{"id":"e5658e9e-e80b-469c-8339-c95b0190b781","cardType":"cloze","front":"A trigonometric identity is true for [ ].","back":"A trigonometric identity is true for all values where both sides are defined.","points":null,"mcqOptions":null},{"id":"1edef8b9-4361-4b6e-b6c9-7e1f070a12f2","cardType":"basic","front":"Why can testing several angles not prove a trigonometric identity?","back":"An identity must hold for infinitely many values, requiring a general argument.","points":2,"mcqOptions":null},{"id":"0e2f1dfa-224b-4550-b1a7-e4049896dc6a","cardType":"cloze","front":"The central Pythagorean identity is $\\text{[...]}$.","back":"The central Pythagorean identity is $\\boxed{\\sin^2 x+\\cos^2 x\\equiv 1}$.","points":null,"mcqOptions":null},{"id":"14b0cee7-c41d-41ab-98a1-4be1df407e07","cardType":"basic","front":"Which identity replaces $1-\\cos^2 x$?","back":"$1-\\cos^2 x\\equiv\\sin^2 x$","points":1,"mcqOptions":null},{"id":"3fcea0b2-4b77-4bad-a843-36c6b09db813","cardType":"cloze","front":"The quotient identity is $\\text{[...]}$, where $\\cos x\\neq0$.","back":"The quotient identity is $\\boxed{\\tan x\\equiv\\frac{\\sin x}{\\cos x}}$, where $\\cos x\\neq0$.","points":null,"mcqOptions":null},{"id":"716f515c-972d-48bd-be91-56e426192521","cardType":"basic","front":"What restriction applies wherever $\\tan x$ appears?","back":"$\\cos x\\neq0$","points":1,"mcqOptions":null},{"id":"d6850ff0-4f67-422d-b4b5-a52ccb733650","cardType":"basic","front":"How is $1+\\tan^2x\\equiv\\sec^2x$ derived from the Pythagorean identity?","back":"Divide every term by $\\cos^2x$.","points":2,"mcqOptions":null},{"id":"fe0f6b94-e50c-4f59-a2f6-ee0429adeec9","cardType":"basic","front":"How is $1+\\cot^2x\\equiv\\csc^2x$ derived from the Pythagorean identity?","back":"Divide every term by $\\sin^2x$.","points":2,"mcqOptions":null},{"id":"78e00081-8898-40a1-ae24-cf0150494e94","cardType":"basic","front":"Which side should usually begin an identity proof?","back":"The more complicated side.","points":1,"mcqOptions":null},{"id":"a22d0271-04b5-4e06-be23-5900caaecfa0","cardType":"basic","front":"Why should an identity proof transform only one side at a time?","back":"It avoids assuming the identity or hiding an irreversible step.","points":2,"mcqOptions":null},{"id":"8c5189b3-cd64-46cb-8aa8-b1a60c4df26a","cardType":"basic","front":"What is a dependable first move when $\\tan x$, $\\sec x$, $\\cot x$ or $\\csc x$ are mixed?","back":"Rewrite them using $\\sin x$ and $\\cos x$.","points":1,"mcqOptions":null},{"id":"86f04abb-c596-496c-95e4-1f9fcb305442","cardType":"basic","front":"What conjugate should multiply $1+\\sin x$?","back":"$1-\\sin x$","points":1,"mcqOptions":null},{"id":"b5af5c2c-c7f1-4992-994a-65f4a4491c49","cardType":"basic","front":"Why is multiplying by the conjugate useful for $(1+\\sin x)(1-\\sin x)$?","back":"It gives $1-\\sin^2x\\equiv\\cos^2x$.","points":2,"mcqOptions":null},{"id":"ceb6e65c-a7b6-4197-9d9a-705d7a2fe374","cardType":"basic","front":"What results after combining $\\frac1{1-\\sin x}-\\frac1{1+\\sin x}$ into one fraction?","back":"$\\frac{2\\sin x}{1-\\sin^2x}$","points":2,"mcqOptions":null},{"id":"1c86d226-480a-455a-ba5d-46f78be63cfc","cardType":"cloze","front":"The sine compound-angle identity is $\\tex