Expand \(\left(z+\frac{1}{z}\right)^{4}\left(z-\frac{1}{z}\right)^{2}\) and, by substituting \(z=\cos \theta+\mathrm{i} \sin \theta\), find integers \(p, q, r, s\) such that
\(64 \sin ^{2} \theta \cos ^{4} \theta=p+q \cos 2 \theta+r \cos 4 \theta+s \cos 6 \theta\)

Using the substitution \(x=2 \cos \theta\), show that
\(\int_{1}^{2} x^{4} \sqrt{ }\left(4-x^{2}\right) \mathrm{d} x=\frac{4}{3} \pi+\sqrt{ } 3\)