As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem:
Working this out using sum-of-angle identities from trigonometry is laborious (see §3.13 for details). However, using Euler's identity, De Moivre's theorem simply ``falls out'':
Moreover, by the power of the method used to show the result, can be any real number, not just an integer.