Let \(I _ { n } = \int _ { 1 } ^ { \sqrt { } 2 } \left( x ^ { 2 } - 1 \right) ^ { n } \mathrm {~d} x\).
- Show that, for \(n \geqslant 1\),
$$( 2 n + 1 ) I _ { n } = \sqrt { } 2 - 2 n I _ { n - 1 } .$$
- Using the substitution \(x = \sec \theta\), show that
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { 2 n + 1 } \theta \sec \theta \mathrm {~d} \theta$$
- Deduce the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 7 } \theta } { \cos ^ { 8 } \theta } \mathrm {~d} \theta$$
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