De Moivre's Theorem relates a complex number to it's polar form. It states

Proof: We can prove De Moivre's Theorem using Taylor series.

We separate the Taylor series into real and complex parts. The real part is the Taylor series fornd the imaginary part is the Taylor series for

This becomes (1) on recognizing that on the right hand side we have expressions for the Taylor series ofandrespectively.

De Moivre's theorem can be used to expressorin terms of powers of orrespectively. To do this we replacewithobtaining

Buthence(2)

Suppose then we want to find expressions forand

We now use thatandobtaining

Now separate the real and imaginary components on both sides, obtaining the two equations

and

Substituteinto the first of these two:

ie

Substituteinto the second equation: