I know it is called Niven's Theorem, but I don't know how to prove it. I am looking for a clue or a direction for where to begin. Here is what I do know:
$\cos{\theta^ {\circ}}$ can be expanded to some long expression of $\sin1^{\circ}$ and $\cos1^{\circ}$ multiplied, added, and subtracted from one another, but only for $\theta \in \mathbb{Z}$. So if this expression is rational and $\theta$ is an integer, it is a fitting solution.
I also know $\cos0^{\circ}=1$, $\cos60^{\circ}=\frac{1}{2}$, $\cos90^{\circ}=0$, and it also works for the negatives of these, and I can add $360^{\circ}$ as many times as I want.
Any hint or tip?
Thanks in advance for help.