 For formula_2 and formula_3, de Moivre's formula asserts that
formula_4
or equivalently that
formula_5
In this example, it is easy to check the validity of the equation by multiplying out the left side.De Moivre's formula is a precursor to Euler's formula
formula_6
which establishes the fundamental relationship between the trigonometric functions and the complex exponential function.One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers

since Euler's formula implies that the left side is equal to formula_8 while the right side is equal to 

The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there.