Упрощение алгебраических выражений. Задача 2

ОТВЕТ: \(\dfrac{{2xy}}{{{x^2} + {y^2}}}\).

\(\left( {\dfrac{{x + y}}{{x-y}}-\dfrac{{x-y}}{{x + y}}} \right){\left( {\dfrac{{x + y}}{{x-y}} + \dfrac{{x-y}}{{x + y}}} \right)^{-1}} = \)

\( = \dfrac{{{x^2} + 2xy + {y^2}-{x^2} + 2xy-{y^2}}}{{\left( {x-y} \right)\left( {x + y} \right)}} \cdot {\left( {\dfrac{{{x^2} + 2xy + {y^2} + {x^2}-2xy + {y^2}}}{{\left( {x-y} \right)\left( {x + y} \right)}}} \right)^{-1}} = \)

\( = \dfrac{{4xy}}{{\left( {x-y} \right)\left( {x + y} \right)}} \cdot \dfrac{{\left( {x-y} \right)\left( {x + y} \right)}}{{2{x^2} + 2{y^2}}} = \dfrac{{4xy}}{{2\left( {{x^2} + {y^2}} \right)}} = \dfrac{{2xy}}{{{x^2} + {y^2}}}.\)

Ответ:  \(\dfrac{{2xy}}{{{x^2} + {y^2}}}.\)