Question 10 - A-Level Maths

Use de Moivre's theorem to show that $$\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta$$
Hence explain why the roots of the equation $16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0$ are $x = \pm \sin \frac { 1 } { 5 } \pi$ and $x = \pm \sin \frac { 2 } { 5 } \pi$.
Without using a calculator, find the exact values of $$\sin \frac { 1 } { 5 } \pi \sin \frac { 2 } { 5 } \pi \sin \frac { 3 } { 5 } \pi \sin \frac { 4 } { 5 } \pi \quad \text { and } \quad \sin ^ { 2 } \left( \frac { 1 } { 5 } \pi \right) + \sin ^ { 2 } \left( \frac { 2 } { 5 } \pi \right) .$$