Find all rational numbers $a$ and integers $b$ such that $\tan{a\pi}=\sqrt{\frac{b+2}{b}}.$

Question

Find all rational numbers $a$ and integers $b$ such that $\tan{a\pi}=\sqrt{\frac{b+2}{b}}$.

Answer 1

We write $\cos2a\pi = \frac{1 - \tan^2 a\pi}{1 + \tan^2 a\pi} = -\frac 1 {b + 1}$ which is a rational number.

It is well known that this can happen only for $\frac 1{b + 1} = 0, \pm \frac 1 2, \pm 1$.