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

$$\cos^7\theta=a\cos7\theta+b\cos5\theta+c\cos3\theta+d\cos\theta,$$

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

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

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