An important property of complex numbers is the fact that two complex numbers are only equal if the real parts are equal and the imaginary parts are equal, ie:
if a + bi = c + di, then:
a=c
and
b=d
Proof:
a + bi = c + di
=> (a + bi) - (c + di) = 0
=> (a - c) + (b - d)i = 0
=> (a - c) = 0 and (b - d) = 0
=> a = c and b = d
This is useful in using De Moivre's Theorem to find Cos(nx) in terms of Cos(x), among other things (it can also be used to find the square root of a complex number)