De Moivre's formula
In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and integer n it holds that
where i is the imaginary unit (i2 = −1). While the formula was named after de Moivre, he never stated it in his works. The expression cos(x) + i sin(x) is sometimes abbreviated to cis(x).
The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x).
As written, the formula is not valid for non-integer powers n. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex numbers z such that zn = 1.
Although historically proven earlier, de Moivre's formula can easily be derived from Euler's formula
and the exponential law for integer powers
Then, by Euler's formula,
The truth of de Moivre's theorem can be established by mathematical induction for natural numbers, and extended to all integers from there. For an integer n, call the following statement S(n):
...
Wikipedia