De Moivre's Theorem

De Moivre’s Theorem is a fundamental statement of complex analysis:

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

Extension of Euler's formula

De Moivre's formula is a trivial extension of Euler's formula:

$e^{ix}=\cos x+i\sin x\,$

Because

$\left(e^{ix}\right)^{n}=e^{inx}\,$

Therefore, from Euler's formula:

$e^{i(nx)}=\cos(nx)+i\sin(nx)\,$