De Moivre's Formula/Negative Integer Index

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

Then:
 * $\forall n \in \Z_{\le 0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$

Proof
Let $n = 0$.

Then:

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

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

Thus: