Let $n$ be a positive integer. Can we precisely solve the equation $\sin(x) = n\cos(x)$ in $x$?
For $n=1$, we get $x=\pi/4$.
Let $n$ be a positive integer. Can we precisely solve the equation $\sin(x) = n\cos(x)$ in $x$?
For $n=1$, we get $x=\pi/4$.
The only rational values of $\tan(x)$ or $\cot(x)$ for rational $x$ in degrees are $0$ and $\pm 1$. See for instance
Olmsted, J. M. H.; Discussions and Notes: Rational Values of Trigonometric Functions. Amer. Math. Monthly 52 (1945), no. 9, 507–508.
From this you cannot expect that your equation has any other nice solutions.