\(A=\left(\dfrac{x}{y^2-xy}+\dfrac{y}{x^2-xy}\right):\left(\dfrac{x^2+y^2}{xy^2+x^2y}\right)\)
\(A=\left(\dfrac{x}{y\left(y-x\right)}+\dfrac{y}{x\left(x-y\right)}\right):\left(\dfrac{x^2+y^2}{xy\left(x+y\right)}\right)\\ \)
\(x,y\ne0;\left|y\right|\ne\left|x\right|\)
\(\)\(A=\left(\dfrac{x}{y\left(y-x\right)}+\dfrac{y}{x\left(x-y\right)}\right).\dfrac{xy\left(x+y\right)}{x^2+y^2}\)
\(A=\left(\dfrac{x}{y\left(y-x\right)}.\dfrac{xy}{x^2+y^2}+\dfrac{y}{x\left(x-y\right)}.\dfrac{xy}{x^2+y^2}\right)\left(x+y\right)\)
\(A=\left(\dfrac{x^2}{\left(y-x\right)\left(x^2+y^2\right)}+\dfrac{y^2}{\left(x-y\right)\left(x^2+y^2\right)}\right)\left(x+y\right)\)
\(A=\dfrac{\left(x-y\right)\left(x+y\right)}{\left(y-x\right)\left(x^2+y^2\right)}\left(x+y\right)=\dfrac{-\left(x+y\right)^2}{x^2+y^2}\)
\(\)
\(\left(\dfrac{x}{-y\left(x-y\right)}+\dfrac{y}{x\left(x-y\right)}\right).\dfrac{xy\left(x+y\right)}{x^2+y^2}\)
\(=\dfrac{x^2-y^2}{-xy\left(x-y\right)}.\dfrac{xy\left(x+y\right)}{x^2+y^2}\)
\(=\dfrac{\left(x-y\right)\left(x+y\right)}{-xy\left(x-y\right)}.\dfrac{xy\left(x+y\right)}{x^2+y^2}\)
\(=\dfrac{\left(x+y\right)^2}{-x^2-y^2}\)
