Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) values for $x$, e.g. $\tan(\frac{\pi}{4})=1$.
Question 1: Does the equation $\tan(x)=y$ have any non-zero solution such that both $x,y$ are rational numbers? In the other words are there natural numbers $m,n,p,q$ such that $\tan(\frac{m}{n})=\frac{p}{q}$? If yes, is the number of such solutions infinite?
Question 2: Does the equation $\tan(x)=y$ have any non-zero solution such that $x$ is an algebraic number and $y$ is rational? If yes, is the number of such solutions infinite?