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\alpha cotα−tanα≡2cot2αfor 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\alpha cot2α−tan2α≡4cot2αcsc2αA particular resonance condition occurs when the operating phase ϕ \phi\,ϕ satisfies the equation
4cot2ϕcsc2ϕ=15tan2ϕ 4 \cot 2\phi \csc 2\phi = 15 \tan^2 \phi 4cot2ϕcsc2ϕ=15tan2ϕSolve this equation for −π2<ϕ<π2\displaystyle -\frac{\pi}{2} < \phi < \frac{\pi}{2}−2π<ϕ<2π, giving your answers to 2 decimal places.