I have a slightly complicated function $f(g) = 2 \arccos\left(\frac{1}{2}(-1+2\cos(4g)+ \cos^2(4g))\right)$.
How do I show that for $g$ which is a rational multiple of $\pi$ excluding the set $\pi/8\mathbb{Z} $, $f(g)$ is not a rational multiple of $\pi$?
More generally what $g$ (assuming it is a rational multiple of $\pi$) should I exclude so that $f(g)$ is not a rational multiple of $\pi$ (I believe the condition above is necessary and sufficient).