Does $\cos (\pi/5)$ belong to $\mathbb{Q} (\sin(\pi/5))$? I need to know if $$\cos(\pi/5) \in \mathbb{Q} (\sin(\pi/5))?$$
I can compute explicitly such $\cos$ and $\sin$, but I have some difficulties how to deduce from this an answer.

Hint : show that $c=\frac{3}{2}-2s^2$, where $c=\cos(\frac{\pi}{5})$
and $s=\sin(\frac{\pi}{5})$. You can check it using Euler's identities for example.
