Prove that for $x\neq 1, 0<y<\pi/2$ the system $u=\sin y/(x-1) ,v=x\tan y$ define a system of curvilinear coordinates.
So this amounts to showing that the transformation is injective. According to solution it says for this [question] it is only necessary to prove that the coordinate curves have different directions and then proceeds to show that the derivatives $dy/dx$ for any $u$ and $v$ are not equal.
Why does this work exactly? Does doing this only work for this particular question or does it work generally?
Also any alternative approach is welcome.