In the analysis of a resonant electronic circuit, the phase shift α \alpha\,α across a specific component is related to the impedance through a series of trigonometric relationships.
Prove that cotα−tanα≡2cot2α\cot \alpha - \tan \alpha \equiv 2 \cot 2\alphacotα−tanα≡2cot2α for α≠nπ2,n∈Z\displaystyle \alpha \neq \frac{n\pi}{2}, n \in \mathbb{Z}α=2nπ,n∈Z.
Using the identity in part (a), or otherwise, establish that cot2α−tan2α≡4cot2αcsc2α\cot^2 \alpha - \tan^2 \alpha \equiv 4 \cot 2\alpha \csc 2\alphacot2α−tan2α≡4cot2αcsc2α
A particular resonance condition occurs when the operating phase ϕ \phi\,ϕ satisfies the equation 4cot2ϕcsc2ϕ=15tan2ϕ4 \cot 2\phi \csc 2\phi = 15 \tan^2 \phi4cot2ϕcsc2ϕ=15tan2ϕ Solve this equation for −π2<ϕ<π2\displaystyle -\frac{\pi}{2} < \phi < \frac{\pi}{2}−2π<ϕ<2π, 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.