Question 4 - A-Level Maths

The complex number $z = \mathrm { e } ^ { \mathrm { i } \theta }$, where $\theta$ is real.
1. Show that
$$z ^ { n } + \frac { 1 } { z ^ { n } } \equiv 2 \cos n \theta$$ 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, making your reasoning clear, determine all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval $0 \leqslant \theta < 2 \pi$