Rational $a$ such that $\sin(a\pi)$ or $\tan(a\pi)$ have the form $\pm\sqrt{a_1}\pm\sqrt{a_2}\pm\cdots\pm\sqrt{a_n}$ for rational $a_i$ I'm interested in simple values of trigonometric functions, e.g., $\sin 30^{\circ} = 1/2$ or $\sin 18^{\circ} = \frac{\sqrt{5}-1}{4}$.

Let $S$ be the set of real numbers that can be written of the form $\pm \sqrt{a_1} \pm \sqrt{a_2} \cdots \pm \sqrt{a_k}$ (here $a_i \in \mathbb{Q}$).

*

*Can we determine all $a \in \mathbb{Q}$ that satisfies $\sin(a \pi) \in S$?

*Can we determine all $a \in \mathbb{Q}$ that satisfies $\tan(a \pi) \in S$?


I've heard of this, but here nested square roots are allowed.
 A: Most probably this is the complete list (between 0 and 90 degrees):
$$\begin{array}{|c|c|c|} \hline
                   &\sin           &\tan\\  \hline
0^{\circ}         &0            &0\\   \hline
7.5^{\circ}         &            &(\sqrt{2}-1)(\sqrt{3}-\sqrt{2})\\   \hline
15^{\circ}         &(\sqrt{6}-\sqrt{2})/4            &2-\sqrt{3}\\   \hline
18^{\circ}         &(\sqrt{5}-1)/4            &\\   \hline
22.5^{\circ}         &            &\sqrt{2}-1\\   \hline
30^{\circ}         &1/2            &\sqrt{3}/3\\   \hline
37.5^{\circ}         &            &(\sqrt{2}+1)(\sqrt{3}-\sqrt{2})\\   \hline
45^{\circ}         &\sqrt{2}/2            &1\\   \hline
52.5^{\circ}         &            &(\sqrt{2}-1)(\sqrt{3}+\sqrt{2})\\   \hline
54^{\circ}         &(\sqrt{5}+1)/4            &\\   \hline
60^{\circ}         &\sqrt{3}/2            &\sqrt{3}\\   \hline
67.5^{\circ}         &            &\sqrt{2}+1\\   \hline
75^{\circ}         &(\sqrt{6}+\sqrt{2})/4            &2+\sqrt{3}\\   \hline
82.5^{\circ}         &            &(\sqrt{2}+1)(\sqrt{3}+\sqrt{2})\\   \hline
90^{\circ}         &1            &\\   \hline
\end{array}$$
Let's prove that $\sin$ and $\tan$ of other angles are not in $S$.
Since $\cos(k \theta)$ is a polynomial of $\cos(\theta)$ for $k \in \mathbb{N}$, if $\cos(\theta) \in S$, $\cos(k \theta) \in S$.
This observation significantly reduces the number of angles we need to check;
for example, once we verify that $\cos(\frac{1}{20} \times 2\pi) \notin S$, we don't need to check angles $\frac{a}{b} \times 2 \pi$ such that $a$ and $b$ are coprime and $b$ is a multiple of $20$. $\tan$ can be handled similarly by using the fact that if $\tan(\theta) \in S$, $\cos(2 \theta) \in S$.
Furthermore, from Galois theory we know which angles can be expressed with (possibly nested) square roots. By combining these results, it suffices to verify that the following values are not in $S$: $\cos(\frac{2\pi}{15}), \cos(\frac{2\pi}{16}), \cos(\frac{2\pi}{20}), \tan(\frac{2\pi}{5})$, and $\cos(\frac{2\pi}{p})$ for Fermat Primes larger than $5$. It's relatively easy to verify the first four.
The conclusion is that, if we can prove the following, the list above is complete.

For all Fermat Prime $p \geq 17$, $\cos(\frac{2 \pi}{p}) \notin S$.

