\(\left\{{}\begin{matrix}\dfrac{1}{\sqrt{x}}+\sqrt{1+\dfrac{1}{y}}=2\left(1\right)\\\dfrac{1}{\sqrt{y}}+\sqrt{1+\dfrac{1}{x}}=2\left(2\right)\end{matrix}\right.\)\(\left(x;y>0\right)\)
\(\left(1\right)-\left(2\right)\Rightarrow\dfrac{1}{\sqrt{x}}-\sqrt{1+\dfrac{1}{x}}+\sqrt{1+\dfrac{1}{y}}-\dfrac{1}{\sqrt{y}}=0\)
\(\Leftrightarrow\dfrac{\dfrac{1}{x}-1-\dfrac{1}{x}}{\dfrac{1}{\sqrt{x}}+\sqrt{1+\dfrac{1}{x}}}+\dfrac{1+\dfrac{1}{y}-\dfrac{1}{y}}{\sqrt{1+\dfrac{1}{y}}+\dfrac{1}{\sqrt{y}}}=0\)
\(\Leftrightarrow\dfrac{-1}{\dfrac{1}{\sqrt{x}}+\sqrt{1+\dfrac{1}{x}}}+\dfrac{1}{\sqrt{1+\dfrac{1}{y}}+\dfrac{1}{\sqrt{y}}}=0\)
\(\Leftrightarrow\dfrac{1}{\sqrt{x}}+\sqrt{1+\dfrac{1}{x}}=\sqrt{1+\dfrac{1}{y}}+\dfrac{1}{\sqrt{y}}\)
\(\Leftrightarrow\dfrac{1}{\sqrt{x}}-\dfrac{1}{\sqrt{y}}+\sqrt{1+\dfrac{1}{x}}-\sqrt{1+\dfrac{1}{y}}=0\)
\(\Leftrightarrow\dfrac{\dfrac{1}{x}-\dfrac{1}{y}}{\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}}+\dfrac{\dfrac{1}{x}-\dfrac{1}{y}}{\sqrt{1+\dfrac{1}{x}}+\sqrt{1+\dfrac{1}{y}}}=0\)
\(\Leftrightarrow\left(\dfrac{1}{x}-\dfrac{1}{y}\right)\left(\dfrac{1}{\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}}+\dfrac{1}{\sqrt{1+\dfrac{1}{x}}+\sqrt{1+\dfrac{1}{y}}}>0\left(x;y>0\right)\right)=0\)
\(\)\(\Leftrightarrow\dfrac{1}{x}=\dfrac{1}{y}\Leftrightarrow x=y\)
\(\Rightarrow\left(1\right)\Leftrightarrow\dfrac{1}{\sqrt{x}}+\sqrt{1+\dfrac{1}{x}}=2\Leftrightarrow\dfrac{1}{\sqrt{x}}-\dfrac{3}{4}+\sqrt{1+\dfrac{1}{x}}-\dfrac{5}{4}=0\)
\(\Leftrightarrow\dfrac{\dfrac{1}{x}-\dfrac{9}{16}}{\dfrac{1}{\sqrt{x}}+\dfrac{3}{4}}+\dfrac{1+\dfrac{1}{x}-\dfrac{25}{16}}{\sqrt{1+\dfrac{1}{x}}+\dfrac{5}{4}}=0\Leftrightarrow\left(\dfrac{1}{x}-\dfrac{9}{16}\right)\left(\dfrac{1}{\dfrac{1}{\sqrt{x}}+\dfrac{3}{4}}+\dfrac{1}{\dfrac{5}{4}+\sqrt{1+\dfrac{1}{x}}}\right)=0\Leftrightarrow\dfrac{1}{x}=\dfrac{9}{16}\Leftrightarrow x=y=\dfrac{16}{9}\)
