De Moivre's Theorem

De Moivre’s Theorem is a fundamental statement of complex analysis, where i represents the square root of (-1):

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

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:

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