De Moivre's Theorem

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de Moivre's Theorem, named after w:Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that

The formula is important because it connects complex numbers and trigonometry.

Proof by Induction

De Moivre's Theorem can be proved for all real numbers as follows;

Assuming the theorem to be true for n = k

Now Consider n = k + 1:

So if the theorem is true for n = k, it is also true for n = k + 1.

      when n = 1       

The theorem is true for n = 1.

Hence by induction De Moivre's theorem is true when n is a positive integer