Problem

(a) Use de Moivre’s theorem to show that

$$\operatorname{cosec}5\theta=\frac{\operatorname{cosec}^5\theta}{5\operatorname{cosec}^4\theta-20\operatorname{cosec}^2\theta+16}.$$

(b) Hence obtain the roots of the equation

$$x^5-10x^4+40x^2-32=0$$

in the form $\operatorname{cosec}(q\pi)$, where $q$ is rational.