VTRMC 1980 Problem 8

Let $z=x+iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1.$ (a) Find two such complex numbers. (b) Show that $|z^{2n}-1|=2|\sin n\theta|,$ where $z=e^{i\theta}.$ (c) Show that $|z^2n -1|$ is rational for every $n.$