Problem

Let

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

(a) Prove that for $n \ge 2$

$$(n-1)I_n = \tan x\,\sec^{n-2}x + (n-2)I_{n-2}.$$

(b) Hence, showing each step of your working, find the exact value of

$$\int_0^{\pi/4} \sec^6 x\,dx.$$