DeMoivre’s Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for $x \in R$ and $n \in Z ,(\cos x+i \sin x)^{n}=\cos (n x)+i \sin (n x)$
Proof:
This is one proof of De Moivre’s theorem by induction.
If $n>0$, for $n=1$, the case is obviously true.
Assume true for the case $n=k$. Now, the case of $n=k+1$:
$${(\cos x + i\sin x)^{k + 1}} = {(\cos x + i\sin x)^k}(\cos x + i\sin x)\;$$$${(\cos x + i\sin x)^{k + 1}} = [\cos (kx) + i\sin (kx)](\cos x + i\sin x)$$$${(\cos x + i\sin x)^{k + 1}} = \cos (kx)\cos x – \sin (kx) + i[\cos (kx)\sin x + \sin (kx)\cos x]\;$$$${(\cos x + i\sin x)^{k + 1}} = \cos (k + 1)x + i\sin (k + 1)x$$
Therefore, the result is true for all positive integers $n$.
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