Problem

(a) By considering the binomial expansion of $\big(z + \frac{1}{z}\big)^6$, where $z = \cos\theta + \mathrm{i}\sin\theta$, use de Moivre’s theorem to show that

$$\cos^6\theta = a\cos 6\theta + b\cos 4\theta + c\cos 2\theta + d,$$

where $a$, $b$, $c$ and $d$ are constants to be determined.

(b) Find the exact value of $\int_0^{\frac{1}{2}\pi}\cos^6 2x\,\mathrm{d}x$.