de Moivre's formula

Statement

For every real number $x$ and every integer $n$, $$(\cos x + i \sin x)^n = \cos(n x) + i \sin(n x).$$

Remarks

Proof: by Euler's formula, the left side is $(e^{i x})^n = e^{i n x}$, which is the right side. For $n = 2$, comparing real and imaginary parts gives $\cos 2x = \cos^2 x - \sin^2 x$ and $\sin 2x = 2 \sin x \cos x$. For non-integer exponents $n$, the formula fails in general.

Explanations

L1 human

Raising a point of the unit circle to the power $n$ multiplies its angle by $n$. For example, $\cos(\pi/4) + i \sin(\pi/4)$ points at $45$ degrees, and its fourth power points at $180$ degrees, so it is $-1$.