Let \(z=\cos \theta+\mathrm{i} \sin \theta\). Show that
\(z^{n}+\frac{1}{z^{n}}=2 \cos n \theta \quad \text { and } \quad z^{n}-\frac{1}{z^{n}}=2 \mathrm{i} \sin n \theta .\)

By considering \(\left(z-\frac{1}{z}\right)^{4}\left(z+\frac{1}{z}\right)^{2}\), show that
\(\sin ^{4} \theta \cos ^{2} \theta=\frac{1}{32}(\cos 6 \theta-2 \cos 4 \theta-\cos 2 \theta+2) .\)

Hence find the exact value of \(\int_{0}^{\frac{1}{4} \pi} \sin ^{4} \theta \cos ^{2} \theta d \theta\).
[Question 11 is printed on the next page.]