Show that
cos2xcosx−sin2xsinx≡−secx,x≠nπ2, n∈Z \frac{\cos 2x}{\cos x} - \frac{\sin 2x}{\sin x} \equiv -\sec x, \quad x \neq \frac{n\pi}{2}, \; n \in \mathbb{Z} cosxcos2x−sinxsin2x≡−secx,x=2nπ,n∈ZHence solve, for 0<θ<π0 < \theta < \pi0<θ<π,
(cos2θcosθ−sin2θsinθ)2=5−4tanθ \left( \frac{\cos 2\theta}{\cos \theta} - \frac{\sin 2\theta}{\sin \theta} \right)^2 = 5 - 4\tan \theta (cosθcos2θ−sinθsin2θ)2=5−4tanθgiving your answers to 3 significant figures as appropriate.
Using the result from part (a), or otherwise, find the exact value of
∫0π3(cos2xcosx−sin2xsinx)tanx dx \int_{0}^{\frac{\pi}{3}} \left( \frac{\cos 2x}{\cos x} - \frac{\sin 2x}{\sin x} \right) \tan x \, dx ∫03π(cosxcos2x−sinxsin2x)tanxdx317 exam-style questions on OCR (MEI) A Level Maths 1.7 Trigonometry, covering 1.7.1 Solve right-angled triangles, 1.7.2 Definitions of sin, cos and tan for any angle, 1.7.3 Graphs of sin, cos and tan, 1.7.4 Exact values of trig functions (degrees), 1.7.5 Area of a triangle, 1.7.6 Sine and cosine rules, 1.7.7 Identity tan = sin/cos, 1.7.8 Identity sin^2 + cos^2 = 1, 1.7.9 Solve simple trigonometric equations, 1.7.10 Exact values of trig functions (radians) (A-level only), 1.7.11 Inverse trigonometric functions (A-level only), 1.7.12 Radians and degree conversion (A-level only), 1.7.13 Arc length and area of a sector (A-level only), 1.7.14 Small angle approximations (A-level only), 1.7.15 Sec, cosec and cot functions (A-level only), 1.7.16 Graphs of reciprocal trig functions (A-level only), 1.7.17 Pythagorean identities for sec and cosec (A-level only), 1.7.18 Compound angle formulae (A-level only), 1.7.19 Double angle identities (A-level only), 1.7.20 Expressions for a cos θ ± b sin θ (A-level only), 1.7.21 Use identities to solve equations (A-level only), 1.7.22 Proofs involving trigonometric functions (A-level only), 1.7.23 Trig to solve problems in context (A-level only), and 1.7 Trigonometry. Each one has a worked solution and a mark scheme showing where the marks go.