De Moivre's Theorem

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

<math>\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right)\,</math>

Extension of Euler's formula

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

<math>e^{ix} = \cos x + i\sin x\,</math>

Because

<math>\left( e^{ix} \right)^n = e^{inx} \,</math>

Therefore, from Euler's formula:

<math>e^{i(nx)} = \cos(nx) + i\sin(nx)\,</math>