De Moivre’s Theorem

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 $n\in Z,(\cos x+i\sin x{)}^n=\cos (nx)+i\sin (nx)$

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 $n$.