Prove that if $({x+\sqrt{x^2+1}})({y+\sqrt{y^2+1}})=1$ then $x+y=0$

Note $$y+\sqrt{y^2+1}=\sqrt{x^2+1}-x\tag{1}$$ $$x+\sqrt{x^2+1}=\sqrt{y^2+1}-y\tag{2}$$ $(1)+(2)$ $$\Longrightarrow x+y=-(x+y)$$ $$\Longrightarrow x+y=0$$

Hint: Let $x=\sinh a$ and $y=\sinh b$. Then, using the fact that $\cosh^2u-\sinh^2u=1$ and $\sinh u+\cosh u=e^u$, we arrive at $e^{a+b}=1\iff a+b=0$, assuming a and b are reals.
Since $\sinh u$, just like $\sin u$, is an odd function, the proof is complete.