Problem

(a) By considering the binomial expansion of $\left(z + \frac{1}{z}\right)^7$, where $z = \cos\theta + \mathrm{i}\sin\theta$, use de Moivre’s theorem to show that

$$\cos^7\theta = a\cos 7\theta + b\cos 5\theta + c\cos 3\theta + d\cos\theta,$$

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

Let

$$I_n = \int_0^{\frac{1}{4}\pi} \cos^n\theta\,\mathrm{d}\theta.$$

(b) Show that

$$nI_n = 2^{-\frac{1}{2}n} + (n - 1)I_{n - 2}.$$

(c) Using the results given in parts (a) and (b), find the exact value of $I_9$.