How to show this property $$\mathrm{arsinh}\ x + \mathrm{arsinh}\,y = \mathrm{arsinh}\left(x\cdot\sqrt{y^2+1}+y\cdot\sqrt{x^2+1}\right)$$ I tried with log formula $\mathrm{arsinh}\,t = \ln\left(t+\sqrt{t^2+1}\right)$ but, I can't simplify this expression : $$xy + \sqrt{y^2+1}\cdot \sqrt{x^2+1}$$ as $\sqrt{X^2 +1}$...
Thanks for help.
Hint
Take the $\sinh$ of both sides.
Then use the fact that $$\sinh (a+b)=\sinh (a) \cosh (b)+\cosh (a) \sinh (b)$$ compute the result for $a=\sinh ^{-1}(x)$ and $b=\sinh ^{-1}(y)$.