I was working with the inequality, $$\left|\sin^{-1}(x) + \sin^{-1}(y)\right|\le\frac{\pi}{2} , \{|x|,|y|\le1\}.$$
I was trying to find an inequality only in terms of x and y without containing any trigonometric functions. Then after some hit and trial I found following beautiful inequality, $$\biggl|x|x| + y|y| \biggr|\le 1 , \{|x|,|y|\le1\}.$$ But the problem is I couldn't derive this inequality and couldn't show their equivalence. I know they are equivalent by plotting them in Desmos(Function Plotting Program). Can you derive the proof showing the equivalence of these two inequalities ?
Note : The above inequality is the condition for the following relation to satisy, $$\sin^{-1}(x) + \sin^{-1}(y) = \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right)$$ I had thought this formula works for every value of $|x|,|y|\le1$ but turns out it doesn't. It only works when x and y satisfy above inequality.