Question 4 - A-Level Maths

If $z = \cos \theta + \mathrm { i } \sin \theta$, use de Moivre's theorem to prove that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ [3 marks]
Express $\sin ^ { 5 } \theta$ in terms of $\sin 5 \theta , \sin 3 \theta$ and $\sin \theta$
[0pt] [4 marks]
Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta \mathrm {~d} \theta = \frac { 53 } { 480 }$$