(a) Use de Moivre's theorem to show that \(\sin ^{6} \theta=-\frac{1}{32}(\cos 6 \theta-6 \cos 4 \theta+15 \cos 2 \theta-10)\).

It is given that \(\cos ^{6} \theta=\frac{1}{32}(\cos 6 \theta+6 \cos 4 \theta+15 \cos 2 \theta+10)\).
(b) Find the exact value of \(\int_{0}^{\frac{1}{3} \pi}\left(\cos ^{6}\left(\frac{1}{4} x\right)+\sin ^{6}\left(\frac{1}{4} x\right)\right) \mathrm{d} x\).
(c) Express each root of the equation \(16 c^{6}+16\left(1-c^{2}\right)^{3}-13=0\) in the form \(\cos k \pi\), where \(k\) is a rational number.