5 In this question you must show detailed reasoning.
It is given that \(I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta\) for \(n \geq 0\).
- Prove that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geq 2\).
- (a) Evaluate \(I _ { 1 }\).
(b) Use the reduction formula to determine the exact value of \(\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta\).