Problem

$$I_n=\int \cos^n x\,dx \qquad n\ge 0$$

(a) Prove that for

$$I_n=\frac1n\cos^{n-1}x\sin x+\frac{n-1}{n}I_{n-2}$$

(b) Show that for positive even integers $n$

$$\int_0^\pi \cos^n x\,dx =\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}\,\pi$$

(c) Hence determine the exact value of

$$\int_0^\pi \cos^6x\sin^2x\,dx$$