(i) Use de Moivre's theorem to show that
\(\sin 5 \theta=5 \sin \theta-20 \sin ^{3} \theta+16 \sin ^{5} \theta .\)

(ii) 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 \(x= \pm \sin \frac{2}{5} \pi\).

(iii) Without using a calculator, find the exact values of
\(\sin \frac{1}{5} \pi \sin \frac{2}{5} \pi \sin \frac{3}{5} \pi \sin \frac{4}{5} \pi \quad \text { and } \quad \sin ^{2}\left(\frac{1}{5} \pi\right)+\sin ^{2}\left(\frac{2}{5} \pi\right) .\)