Let \(I_n=\int_1^{\sqrt2}(x^2-1)^n\,dx\).

(i) Show that, for \(n\ge1\), \((2n+1)I_n=\sqrt2-2nI_{n-1}\).

(ii) Using the substitution \(x=\sec\theta\), show that \(I_n=\int_0^{\pi/4}\tan^{2n+1}\theta\sec\theta\,d\theta\).

(iii) Deduce the exact value of \(\int_0^{\pi/4}\frac{\sin^7\theta}{\cos^8\theta}\,d\theta\).