Well, it is easy enough to prove that if

**x+y=0**

then

**(x+sqrt{1+y^2})*(y+sqrt{1+x^2})=1**** **

Just substitute y = -x in the messy formula, and do the multiplication, and simplify.

Unfortunately, that doesn't prove that this is the only solution.

Squaring both sides of the equation didn't seem to help.

I guess that some more clever substitution is called for, but I'm not finding it. I tried c = x + y, with a hope of proving that c = 0, but substituting y = c-x didn't seem to help either.

This problem is a difficulty 2? I will be very interested to see the solution.