6. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.
- Use de Moivre's theorem to show that
$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
where \(n\) is a positive integer.
- Show that
$$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
- Hence find all the solutions of
$$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$
in the interval \(0 \leqslant \theta < 2 \pi\)