Prove that tanϕ+cotϕ≡2csc2ϕ\tan \phi + \cot \phi \equiv 2 \csc 2\phitanϕ+cotϕ≡2csc2ϕ for ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}ϕ=2nπ,n∈Z.
Using the identity in part (a), or otherwise, prove that cot2ϕ−tan2ϕ≡4cot2ϕcsc2ϕ\cot^2 \phi - \tan^2 \phi \equiv 4 \cot 2\phi \csc 2\phicot2ϕ−tan2ϕ≡4cot2ϕcsc2ϕ for ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}ϕ=2nπ,n∈Z.
Hence solve, for −π2<α<π2-\frac{\pi}{2} < \alpha < \frac{\pi}{2}−2π<α<2π, 4cot2αcsc2α=3tan2α+14 \cot 2\alpha \csc 2\alpha = 3 \tan^2 \alpha + 14cot2αcsc2α=3tan2α+1 giving your answers to 2 decimal places.
Practise Edexcel A Level Maths 7.5 Proving Trigonometric Identities with exam-style questions for A Level Maths. 27 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.