For a complex number
The roots can be found using the formula
Trig identities can be found by equating complex numbers and using de moivre's theorem. The examples below are shown for n=2 but the process is the same for any n.
Using de moivre's theorem to equate
Equating real and imaginary parts
To find the identity for , start with , and raise to the power of 2
Substituting in for the pairs of