De Moivre's Formula/Integer Index/Corollary

Corollary to De Moivre's Formula: Positive Integer Index
Then:
 * $\forall n \in \Z: \paren {\cos x + i \sin x}^n = \map \cos {n x} + i \, \map \sin {n x}$

Proof
$\cos x + i \sin x$ is a complex number expressed in polar form $\polar {r, \theta}$ whose complex modulus is $1$ and whose argument is $x$.

From De Moivre's Formula: Integer Index:
 * $\forall n \in \Z: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$

The result follows by setting $r = 1$.