1. (a) Show that

$$\left( z + \frac { 1 } { z } \right) ^ { 3 } \left( z - \frac { 1 } { z } \right) ^ { 3 } = z ^ { 6 } - \frac { 1 } { z ^ { 6 } } - k \left( z ^ { 2 } - \frac { 1 } { z ^ { 2 } } \right)$$ where \(k\) is a constant to be found. Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), where \(\theta\) is real,
(b) show that

1. \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\)
2. \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 i \sin n \theta\)
   (c) Hence show that $$\cos ^ { 3 } \theta \sin ^ { 3 } \theta = \frac { 1 } { 32 } \quad ( 3 \sin 2 \theta - \sin 6 \theta )$$ (d) Find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 8 } } \cos ^ { 3 } \theta \sin ^ { 3 } \theta d \theta$$ \includegraphics[max width=\textwidth, alt={}, center]{b197811e-1df5-4937-b0d8-f98f82412c76-32_227_148_2524_1797}