De Moivre's Formula/Negative Integer Index

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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 $n = -m$

$\blacksquare$


Sources