De Moivre's Formula/Positive Integer Index/Proof 2

Theorem

 * $r \left({\cos x + i \sin x}\right)^n = r^n \left({\cos \left({n x}\right) + i \sin \left({n x}\right)}\right)$, for $\begin{cases} n = 1, 2, 3, \ldots \\ x \in \R \end{cases}$

Proof
From Product of Complex Numbers in Polar Form: General Result:
 * $z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \left({\cos \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right) + i \sin \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right)}\right)$

Setting $z_1 = z_2 = \cdots = z_n$ gives the result.