5.
$$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x , \quad 0 < x < \frac { \pi } { 2 } , \quad n \geqslant 0$$
- Show that, for \(n \geqslant 2\)
$$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { 1 } { n - 1 } \cot x \operatorname { cosec } ^ { n - 2 } x$$
- Hence, or otherwise, find
$$\int \operatorname { cosec } ^ { 4 } x \mathrm {~d} x$$
giving your answer in terms of \(\cot x\).