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.
Doing it backwards
