I'd like to know the way to prove the transformation is bijective. For example,
(1) the transformation $x=\frac{\sin u}{\cos v}$ and $y=\frac{\sin v}{\cos u}$ is bigective from $\{(u,v): u>0, v> 0, u+v < \frac{\pi}{2}\}$ to $\{(x,y): 0< x,y <1\}.$
(2) the transformation $x=\frac{\cos v}{\cos u}$ and $y=\frac{\sin u}{\sin v}$ is bigective from $\{(u,v): 0< u < v < \frac{\pi}{2}\}$ to $\{(x,y): 0< x,y < 1\}.$
Note that both has Jacobian determinant $J(u,v)=1-x^2y^2.$
Please let me know if you have any comment for this question. Thanks in advance!
Just prove your Jacobian determinant is nonzero for all values of $x,y$. This is due to the inverse function theorem.