Question 7 - A-Level Maths

Prove that if $z = \mathrm { e } ^ { \mathrm { i } \theta }$, then $z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$.
Express $\cos ^ { 6 } \theta$ in terms of cosines of multiples of $\theta$, and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 6 } \theta \mathrm {~d} \theta$$