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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.

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