Complex Addition: $(i+\sqrt3/2)^{200} + (i-\sqrt3/2)^{200}$ Solving a question, I need to find the value of following in between the solution.
$$\left(\frac{i+\sqrt3}{2}\right)^{200} + \left(\frac{i-\sqrt3}{2}\right)^{200}$$
The only useful thing I got was 

$$\left(\frac{i+\sqrt3}{2}\right)^{100}\left(\frac{i-\sqrt3}{2}\right)^{100} = 1$$

Which might be useful to complete the square.

If  $a=\dfrac{i+\sqrt3}2, b=\dfrac{\sqrt3-i}2$
$a+b=\sqrt3,ab=1\implies a^2+b^2=(a+b)^2-2ab=1$
So, $a,b$ are the roots of $t^2-\sqrt3t+1=0$
$\implies(t^2+1)^2=(\sqrt3t)^2$
$\iff t^4-t^2+1=0\implies(t^2+1)(t^4-t^2+1)=0\iff t^6=-1$
$\implies t^{2+6(2n+1)}=(t^6)^{2n+1}\cdot t^2=-t^2$
$\implies a^{12n+8}+b^{12n+8}=-(a^2+b^2)=-1$
