(a) By considering the binomial expansion of \(\left(z+z^{-1}\right)^{4}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), use de Moivre's theorem to show that \(\cos ^{4} \theta=\frac{1}{8}(\cos 4 \theta+4 \cos 2 \theta+3)\).
(b) Use the substitution \(x=\sin \theta\) to find the exact value of \(\int_{0}^{\frac{1}{2}}\left(1-x^{2}\right)^{\frac{3}{2}} \mathrm{~d} x\).