8 Given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\) and \(n\) is a positive integer, show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ Hence express \(\sin ^ { 6 } \theta\) in the form $$p \cos 6 \theta + q \cos 4 \theta + r \cos 2 \theta + s$$ where the constants \(p , q , r , s\) are to be determined. Hence find the mean value of \(\sin ^ { 6 } \theta\) with respect to \(\theta\) over the interval \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).