(a) Use the trigonometric identities
$$
\begin{aligned}
\cos (A+B) &=\cos A \cos B-\sin A \sin B \\
\sin (A+B) &=\cos A \sin B+\sin A \cos B
\end{aligned}
$$
to show that
$$
(\cos A+i \sin A)(\cos B+i \sin B)=\cos (A+B)+i \sin (A+B)
$$
(b) Apply (a) repeatedly to show that if \(n\) is a positive integer then
$$
\prod_{k=1}^{n}\left(\cos A_{k}+i \sin A_{k}\right)=\cos
\left(A_{1}+A_{2}+\cdots+A_{n}\right)+i \sin
\left(A_{1}+A_{2}+\cdots+A_{n}\right)
$$
(c) Infer from (b) that if \(n\) is a positive integer then
$$
(\cos \theta+i \sin \theta)^{n}=\cos n \theta+i \sin n \theta
$$
(d) Show that (A) also holds if \(n=0\) or a negative integer. HINT: Verify by
direct calculation that
$$
(\cos \theta+i \sin \theta)^{-1}=(\cos \theta-i \sin \theta)
$$
Then replace \(\theta\) by \(-\theta\) in \((A) .\)
(e) Now suppose \(n\) is a positive integer. Infer from (A) that if
$$
z_{k}=\cos \left(\frac{2 k \pi}{n}\right)+i \sin \left(\frac{2 k
\pi}{n}\right), \quad k=0,1, \ldots, n-1
$$
and
$$
\zeta_{k}=\cos \left(\frac{(2 k+1) \pi}{n}\right)+i \sin \left(\frac{(2 k+1)
\pi}{n}\right), \quad k=0,1, \ldots, n-1,
$$
then
$$
z_{k}^{n}=1 \quad \text { and } \quad \zeta_{k}^{n}=-1, \quad k=0,1, \ldots,
n-1 .
$$
(Why don't we also consider other integer values for \(k ?)\)
(f) Let \(\rho\) be a positive number. Use
(e) to show that
$$
z^{n}-\rho=\left(z-\rho^{1 / n} z_{0}\right)\left(z-\rho^{1 / n} z_{1}\right)
\cdots\left(z-\rho^{1 / n} z_{n-1}\right)
$$
and
$$
z^{n}+\rho=\left(z-\rho^{1 / n} \zeta_{0}\right)\left(z-\rho^{1 / n}
\zeta_{1}\right) \cdots\left(z-\rho^{1 / n} \zeta_{n-1}\right)
$$
