Question 8 - A-Level Maths

By expressing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$, show that $$\sin ^ { 6 } \theta \equiv - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )$$
Replace $\theta$ by ( $\frac { 1 } { 2 } \pi - \theta$ ) in the identity in part (i) to obtain a similar identity for $\cos ^ { 6 } \theta$.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sin ^ { 6 } \theta - \cos ^ { 6 } \theta \right) \mathrm { d } \theta$.