De Moivre's Formula/Negative Integer Index

Theorem
Let $z \in \C$ be a complex number expressed in complex form:
 * $z = r \left({\cos x + i \sin x}\right)$

Then:
 * $\forall n \in \Z_{\le 0}: \left({r \left({\cos x + i \sin x}\right)}\right)^n = r^n \left({\cos \left({n x}\right) + i \sin \left({n x}\right)}\right)$

Proof
Let $n = 0$.

Then:

Now let $n \in \Z_{<0}$.

Let $n = -m$ where $m > 0$.

Thus: