Given that cosϕ≠±1\cos \phi \neq \pm 1cosϕ=±1, prove the identity
11−cosϕ+11+cosϕ≡2csc2ϕ \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} \equiv 2 \csc^2 \phi 1−cosϕ1+1+cosϕ1≡2csc2ϕIn a study of robotic arm resonance, the stability factor kkk is defined by the equation
11−cosϕ+11+cosϕ=k \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} = k 1−cosϕ1+1+cosϕ1=kDetermine the set of values of kkk for which this equation has real solutions for ϕ\phiϕ. Fully justify your answer.
Given that ϕ\phiϕ is in the third quadrant (reflex angle between 180∘180^\circ180∘ and 270∘270^\circ270∘) and
11−cosϕ+11+cosϕ=18 \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} = 18 1−cosϕ1+1+cosϕ1=18find the exact value of cotϕ\cot \phicotϕ.