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 \phi1−cosϕ1+1+cosϕ1≡2csc2ϕ
In 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} = k1−cosϕ1+1+cosϕ1=k Determine 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} = 181−cosϕ1+1+cosϕ1=18 find the exact value of cotϕ\cot \phicotϕ.