<s>[INST] $(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1$ is true if and only if $x+y=0$ Prove that $(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1$ is true if and only if $x+y=0$  
I believe x and y could both be 0 as that satisfies the equations. Beyond that, I do not know how to prove this.
 [/INST] By continuous rearranging and squaring:
$$\text{Equation} \Rightarrow x+\sqrt{x^2+1}=\frac{1}{y+\sqrt{y^2+1}} \Rightarrow x+y=\sqrt{y^2+1}-\sqrt{x^2+1} \Rightarrow$$
$$1-xy=\sqrt{x^2y^2+x^2+y^2+1} \Rightarrow (x+y)^2=0.$$
</s>