$\begin{array}{l}(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1\\\Leftrightarrow (x-\sqrt{x^2 +1})(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=x-\sqrt{x^2 +1}\\ \Leftrightarrow [x^2 - (x^2 +1)](y+\sqrt{y^2+1})=x-\sqrt{x^2 +1}\\ \Leftrightarrow -1(y+\sqrt{y^2+1})=x-\sqrt{x^2 +1}\\ \Leftrightarrow y + \sqrt{y^2 + 1} = \sqrt{x^2 +1} - x\\ \text{Tương tự, ta được:}\,\,x + \sqrt{x^2 + 1} = \sqrt{y^2 + 1} - y\\ \text{Cộng vế theo vế ta được:}\\ x + y + \sqrt{x^2 + 1} + \sqrt{y^2 + 1} = \sqrt{x^2 + 1} + \sqrt{y^2 + 1} - x - y\\ \Leftrightarrow 2x + 2y = 0\\ \Leftrightarrow x + y = 0 \quad (đpcm) \end{array}$
