Applying de Moivre's Theorem, (cos nx + i sin nx) = (cos x + sin x)^n, almost the only way to make this equation valid is for all the factors equal to 1. That means cos ø + i sinø = 1 so ø = 2pi z, where z is an integer. But when ø = pi/2 + 2pi z, the cosines drop out and we get the expression i * i^2 * i^3 * i^4 etc. That gives us a repeating pattern of i, -1, -i, 1 depending on the value of n. So this value for ø can only be used when n = 4N where N is an integer.