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:

 $\ds r^0 \paren {\map \cos {0 x} + i \map \sin {0 x} }$ $=$ $\ds 1 \times \paren {\cos 0 + i \sin 0}$ Definition of Zeroth Power $\ds$ $=$ $\ds 1 \paren {1 + i 0}$ Cosine of Zero is One and Sine of Zero is Zero $\ds$ $=$ $\ds 1$ $\ds$ $=$ $\ds \paren {r \paren {\cos x + i \sin x} }^0$ Definition of Zeroth Power

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

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

Thus:

 $\ds \paren {r \paren {\cos x + i \sin x} }^{-m}$ $=$ $\ds \frac 1 {\paren {r \paren {\cos x + i \sin x} }^m}$ $\ds$ $=$ $\ds \frac 1 {r^m \paren {\map \cos {m x} + i \map \sin {m x} } }$ De Moivre's Formula: Positive Integer Index $\ds$ $=$ $\ds r^{-m} \paren {\map \cos {-m x} + i \map \sin {-m x} }$ Definition of Complex Division $\ds$ $=$ $\ds r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ as $n = -m$

$\blacksquare$