De Moivre's Formula/Integer Index/Corollary

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

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

From De Moivre's Formula: Integer Index:
 * $\forall n \in \Z: \left({r \left({\cos x + i \sin x}\right)}\right)^n = r^n \left({\cos \left({n x}\right) + i \sin \left({n x}\right)}\right)$

The result follows by setting $r = 1$.