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 A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.