Problem

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

$$\cos 5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta.$$

(b) Hence obtain the roots of the equation

$$32x^5-40x^3+10x-\sqrt{2}=0$$

in the form $\cos(q\pi)$, where $q$ is a rational number.