Let $a, y \in \mathbb{R}^+$ with $y<a$, and $b, x \in [0, \pi/2]$. Now consider the system of two equations given by
\begin{align} a &= \frac{1}{\tan(x)} + y\\ b & = (x+y) \mod \frac{\pi}{2} \end{align}
where mod with real numbers is defined in the natural way. Now, for fixed, known values $a, b$, I am supposed to determine if there exist an uncountably infinite, countably infinite, or finite set of $x, y$ pairs that satisfy the system of equations. Would anyone happen to know how to go about doing this, saying how they arrived at the answer they did? Thanks!