By expanding the left hand side and then comparing real and imaginary parts, it is possible to derive
useful expressions for cos(*nx*) and sin(*nx*) in terms of sin(*x*) and cos(*x*). Furthermore, one can use the formula to find explicit expressions for the *n*-th roots of unity: complex numbers *z* such that *z ^{n}* = 1.

Abraham de Moivre was a good friend of Newton; in 1698 he wrote that the formula had been known to Newton as early as 1676. It can be derived from (but historically preceded) Euler's formula *e*^{ix} = cos *x* + *i* sin *x*
and the exponential law
(*e*^{ix})^{n} = *e*^{inx} (see exponential function).

De Moivre's formula is actually true more generally than stated above: if *z* and *w* are complex numbers, then (cos *z* + *i* sin *z*)^{w} is a multivalued function while cos (*wz*) + *i* sin (*wz*) is not, and one can state that

- cos (
*wz*) +*i*sin (*wz*)**is one value of**(cos*z*+*i*sin*z*)^{w}.