De Moivre's Theorem

DeMoivre's Theorem is a very useful theorem in the mathematical fields of Complex numbers. It states that:

$\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)$

Proof

This is one proof of DeMoivre's theorem by Mathematical Induction.

For $n=1$ , the case is obviously true.

Assume true for the case $n=k$ .

Now, the case of $n=k+1$ .

Therefore, the result is true for all positive integers $n$ .

The formula holds true when $n=0$ because $\cos(0x)+i\sin (0x)=1+i0=1$ . Since $z^0=1$ , the equation holds true.

If $n<0$ , one must consider $n=-m$ when $m$ is a positive integer.

Therefore:

And thus, the formula proves true for all integral values of $n$ . $\Box$