- (a) Given that \(z = e ^ { i \theta }\), show that
$$z ^ { p } + \frac { 1 } { z ^ { p } } = 2 \cos p \theta$$
where \(p\) is a positive integer.
(b) Given that
$$\cos ^ { 4 } \theta = A \cos 4 \theta + B \cos 2 \theta + C$$
find the values of the constants \(A , B\) and \(C\).
The region \(R\) bounded by the curve with equation \(y = \cos ^ { 2 } x , - \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }\), and the \(x\)-axis is rotated through \(2 \pi\) about the \(x\)-axis.
(c) Find the volume of the solid generated.