3.5 Trigonometry (A-level only)

Given that $cos⁡ϕ≠±1\cos \phi \neq \pm 1$ , prove the identity

\frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} \equiv 2 \csc^2 \phi

[3]

In a study of robotic arm resonance, the stability factor $k$ is defined by the equation

\frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} = k

Determine the set of values of $k$ for which this equation has real solutions for $ϕ\phi$ . Fully justify your answer.

[3]

Given that $ϕ\phi$ is in the third quadrant (reflex angle between $180∘180^\circ$ and $270∘270^\circ$ ) and

\frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} = 18

find the exact value of $cot⁡ϕ\cot \phi$ .

[3]