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)tanxdx137 exam-style questions on Edexcel A Level Maths Trigonometry and Modelling, covering 7.1 Addition Formulae, 7.2 Double Angle Formulae, 7.3 Solving Trigonometric Equations, 7.4 Simplifying a cos x +- b sin x, 7.5 Proving Trigonometric Identities, and 7.6 Modelling with Trigonometric Functions. Each one has a worked solution and a mark scheme showing where the marks go.