$\begin{array}{l} x + y + xy = 1 \Rightarrow x + y + xy + 1 = 2\\ \Rightarrow \left( {x + 1} \right)\left( {y + 1} \right) = 2\\ \sqrt {\dfrac{{2\left( {1 + {y^2}} \right)}}{{\left( {1 + {x^2}} \right)}}} = \sqrt {\dfrac{{2\left( {xy + x + y + {y^2}} \right)}}{{\left( {x + y + xy + {x^2}} \right)}}} \\ = \sqrt {\dfrac{{2\left( {y + x} \right)\left( {y + 1} \right)}}{{\left( {x + y} \right)\left( {x + 1} \right)}}} = \sqrt {\dfrac{{2\left( {y + 1} \right)}}{{\left( {x + 1} \right)}}} = \sqrt {\dfrac{{\left( {x + 1} \right){{\left( {y + 1} \right)}^2}}}{{\left( {x + 1} \right)}}} \\ = y + 1\\ \Rightarrow x\sqrt {\dfrac{{2\left( {1 + {y^2}} \right)}}{{\left( {1 + {x^2}} \right)}}} = x\left( {y + 1} \right)\\ TT:y.\sqrt {\dfrac{{2\left( {1 + {x^2}} \right)}}{{1 + {y^2}}}} = y\left( {x + 1} \right)\\ \sqrt {\dfrac{{\left( {1 + {x^2}} \right)\left( {1 + {y^2}} \right)}}{2}} = \sqrt {\dfrac{{\left( {x + y} \right)\left( {x + 1} \right)\left( {x + y} \right)\left( {y + 1} \right)}}{{\left( {x + 1} \right)\left( {y + 1} \right)}}} \\ = \sqrt {{{\left( {x + y} \right)}^2}} = x + y\\ \Rightarrow S = x\left( {y + 1} \right) + y\left( {x + 1} \right) + x + y = 2\left( {xy + x + y} \right) = 2\\ \end{array}$
