Complex numbers: proof of equality

Can anyone help me prove following equality?

$$\left(\frac{-1+i\sqrt{3}}{2}\right)^n + \left(\frac{-1-i\sqrt{3}}{2}\right)^n$$ $$=2 \iff \frac{n}3\in \mathbb{N}$$
$$=-1 \iff \frac{n}3\not\in \mathbb{N}$$
This is what I've got:
$$\left(\frac{-1+i\sqrt{3}}{2}\right)^n + \left(\frac{-1-i\sqrt{3}}{2}\right)^n$$
$$= \left(\frac{2(cos(-\pi/3)+i sin(-\pi/3))}2\right)^n + \left(\frac{2(cos(\pi/3)+i sin(\pi/3))}2\right)^n$$
$$=[cos(-\pi/3)+isin(-\pi/3)]^n-[cos(\pi/3)+isin(\pi/3)]^n$$
$$=cos(n\pi/3)-isin(n\pi/3)-cos(n\pi/3)-isin(n\pi/3)$$
$$=-2isin(n\pi/3)$$

and if $n/3 \in \mathbb{N}$, I get $n'\pi$ and $sin(n'\pi)=0$ which isn't the result I needed...