7 By considering the binomial expansion of \(\left( z - \frac { 1 } { z } \right) ^ { 6 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(\sin ^ { 6 } \theta\) in the form
$$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$
where \(p , q , r\) and \(s\) are integers to be determined.
Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta\).
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7 By considering the binomial expansion of $\left( z - \frac { 1 } { z } \right) ^ { 6 }$, where $z = \cos \theta + \mathrm { i } \sin \theta$, express $\sin ^ { 6 } \theta$ in the form
$$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$
where $p , q , r$ and $s$ are integers to be determined.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta$.
\hfill \mbox{\textit{CAIE FP1 2013 Q7}}