A limit and a coordinate trigonometric transformation of the interior points of a square into the interior points of a triangle
Solution 1:
Question 1: Setting $u=\pi/2-v$ in your first set of formulas gives $(x,y)=(1,1)$, so that point corresponds to the whole hypotenuse $u+v=\pi/2$.
Question 2: I have no idea. (Luck? Inspiration? Trial and error?)