[tex]\dfrac{1}{(x+y)^{2}} \cdot\left(\dfrac{1}{x^{2}}+\dfrac{1}{y^{2}}\right)+\dfrac{2}{(x+y)^{3}}\cdot\left(\dfrac{1}{x}+\dfrac{1}{y}\right)=\\\\\\=\dfrac{1}{(x+y)^{2}}\cdot\left(\dfrac{x^{2}+y^{2}}{x^{2}y^{2}}+\dfrac{2}{x+y}\cdot\dfrac{x+y}{xy}\right)=\\\\\\=\dfrac{1}{(x+y)^{2}}\cdot\left(\dfrac{x^{2}+y^{2}}{x^{2}y^{2}}+\dfrac{2}{xy}\right)=[/tex]
[tex]=\dfrac{1}{xy(x+y)^{2}}}\cdot(\dfrac{x^{2}+y^{2}}{xy}+2)=\\\\\\=\dfrac{1}{xy(x+y)^{2}}\cdot\left(\frac{x^{2}+y^{2}+2xy}{xy}\right)=\\\\\\=\dfrac{1}{xy(x+y)^{2}}\cdot\dfrac{\left(x+y\right)^{2}}{xy}=\dfrac{1}{x^{2}y^{2}}[/tex]
[tex]Z:\\x,y \neq 0[/tex]
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[tex]\dfrac{1}{(x+y)^{2}} \cdot\left(\dfrac{1}{x^{2}}+\dfrac{1}{y^{2}}\right)+\dfrac{2}{(x+y)^{3}}\cdot\left(\dfrac{1}{x}+\dfrac{1}{y}\right)=\\\\\\=\dfrac{1}{(x+y)^{2}}\cdot\left(\dfrac{x^{2}+y^{2}}{x^{2}y^{2}}+\dfrac{2}{x+y}\cdot\dfrac{x+y}{xy}\right)=\\\\\\=\dfrac{1}{(x+y)^{2}}\cdot\left(\dfrac{x^{2}+y^{2}}{x^{2}y^{2}}+\dfrac{2}{xy}\right)=[/tex]
[tex]=\dfrac{1}{xy(x+y)^{2}}}\cdot(\dfrac{x^{2}+y^{2}}{xy}+2)=\\\\\\=\dfrac{1}{xy(x+y)^{2}}\cdot\left(\frac{x^{2}+y^{2}+2xy}{xy}\right)=\\\\\\=\dfrac{1}{xy(x+y)^{2}}\cdot\dfrac{\left(x+y\right)^{2}}{xy}=\dfrac{1}{x^{2}y^{2}}[/tex]
[tex]Z:\\x,y \neq 0[/tex]