<s>[INST] Prove that a map is Injective How can I prove that $f: (0, \infty) \times (0,\pi) \to \mathbb{R}^2$ where $f(x,y) = (\sinh(x)\sin(y),\cosh(x)\cos(y))$ is injective? 
 [/INST] Suppose $\cosh a\cos b=\cosh c\cos d$ and $\sinh a\sin b=\sinh c\sin d$. Square the two relations:
\begin{gather}
\cosh^2 a\cos^2 b=\cosh^2 c\cos^2 d\\
\sinh^2 a\sin^2 b=\sinh^2 c\sin^2 d
\end{gather}
Sum and subtract the two:
\begin{gather}
\cosh^2 a + \sinh^2 a = \cosh^2 c + \sinh^2 c\\
\cos^2 b - \sin^2 b = \cos^2 d - \sin^2 d
\end{gather}
Therefore
$$
\cosh 2a = \cosh 2c
$$
which means that $2a=2c$ because $\cosh$ is injective in $(0,\infty)$.
Furthermore
$$
\cos 2b = \cos 2d
$$
that is, $2b=2d$ or $2b=2\pi-2d$, from which
$$
b=d\quad\text{or}\quad b=\pi-d
$$
The second possibility, together with $\cosh a\cos b=\cosh c\cos d$, gives $\cos b=\cos(\pi-b)$, that is $\cos b=0$ and, again, $b=d=\pi/2$.
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