I have a hypothesis about regular polygons, but in order to prove or disprove it I need a way to determine whether an expression is rational. Once I boil down my expression the only part that could be irrational is:
$S_N = \cot \frac{\pi}{N} \text{ for } N\in ℕ_1 ∖ \left\{1, 2, 4\right\}$
Is there at least one such $N$ for which $S_N$ is rational? Can it be proven that $S_N$ is never rational for any such $N$? How would I go about proving one or the other?