8 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x\) where \(n\) is a positive integer.
- By writing \(\sec ^ { n } x = \sec ^ { n - 2 } x \sec ^ { 2 } x\), or otherwise, show that
$$( n - 1 ) I _ { n } = ( \sqrt { 2 } ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 } \text { for } n > 1 .$$
- Show that \(I _ { 8 } = \frac { 96 } { 35 }\).
- Prove by induction that \(I _ { 2 n }\) is rational for all values of \(n > 1\).
\section*{END OF QUESTION PAPER}