The following identity is a formula found by Abraham De Moivre (1667 - 1754).
For all real numbers $\phi\in\mathbb R$ and all natural numbers $n\in\mathbb N,$ the formula
$$[\cos(\phi)+i\sin(\phi)]^n=\cos(n\phi)+i\sin(n\phi)$$
holds, where $\cos$ and $\sin$ denote the cosine and sine functions.
In particular, the power $z^n$ of a complex number written in polar coordinates $z=r\exp(i\phi)$ can be calculated by the formula $$z^n=r^n\exp(in\phi).$$
Proofs: 1
