Statement of de Moivre's Theorem is as follows:

If n is an integer, positive or negative, or a rational number, then the value or one of the values of the complex number `(costheta + isintheta)^n` is `(cosntheta + isinntheta)`.

This theorem is often used in manipulation of complex numbers. There are numerous applications of de Moivre’s theorem, for example, proving trigonometric identities, finding the nth roots of unity and solving polynomial equations with complex roots.

Some useful trigonometric identities obtained as corollary to this theorem are:

1. `(costheta + isintheta)^n = (cosntheta + isinntheta)`

2. `(costheta - isintheta)^n = (cosntheta - isinntheta)`

3. `(costheta + isintheta)^(-n) = (cosntheta - isinntheta)`

De-Moivre 's Theorem

Let `z=r(cos(theta)+isin(theta))!=0` be a complex number and n is

an integer.Then

`z^n=r^n(cos(ntheta)+isin(ntheta))` .