7 Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), show that
- \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\).
- \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
Hence show that
$$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta )$$
Find a similar expression for \(\cos ^ { 6 } \theta\), and hence express \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in the fom \(a \cos 2 \theta + b \cos 6 \theta\).