Proof of De Moivre's Theorem
Theorem
If $z = r \text{cis} \theta$, then $z^n = r^n \text{cis} n\theta$ holds for every natural number $n$.
- $\text{cis} \theta: = \cos \theta + i \sin \theta$
Proof
Let us use mathematical induction.
For $n=1$ it is trivial, and assuming it also holds for $n=k$, we have $$ z^{k+1} = z z^k = (r \text{cis} \theta)(r^k \text{cis} k\theta) $$ Meanwhile, since $z_1 z_2 = r_1 r_2 \text{cis} (\theta_1 + \theta_2)$, $$ z^{k+1} = r^{k+1} \text{cis} (k+1)\theta $$
Since it holds for $n=k+1$ whenever it holds for $n=k$, the given formula holds for all natural numbers.
■