a) Với \(P = \dfrac{{xy}}{{x - y}}\)
Ta có:
\(\dfrac{{xP}}{{x + P}} - \dfrac{{yP}}{{y - P}}\)
\( = \dfrac{{\dfrac{{{x^2}y}}{{x - y}}}}{{x + \dfrac{{xy}}{{x - y}}}} - \dfrac{{\dfrac{{x{y^2}}}{{x - y}}}}{{y - \dfrac{{xy}}{{x - y}}}}\)
\( = \dfrac{{\dfrac{{{x^2}y}}{{x - y}}}}{{\dfrac{{x\left( {x - y} \right) + xy}}{{x - y}}}} - \dfrac{{\dfrac{{x{y^2}}}{{x - y}}}}{{\dfrac{{y\left( {x - y} \right) - xy}}{{x - y}}}}\)
\( = \dfrac{{\dfrac{{{x^2}y}}{{x - y}}}}{{\dfrac{{{x^2} - xy + xy}}{{x - y}}}} - \dfrac{{\dfrac{{x{y^2}}}{{x - y}}}}{{\dfrac{{xy - {y^2} - xy}}{{x - y}}}} \)
\(= \dfrac{{\dfrac{{{x^2}y}}{{x - y}}}}{{\dfrac{{{x^2}}}{{x - y}}}} - \dfrac{{\dfrac{{x{y^2}}}{{x - y}}}}{{\dfrac{{ - {y^2}}}{{x - y}}}}\)
\( = \left( {\dfrac{{{x^2}y}}{{x - y}}.\dfrac{{x - y}}{{{x^2}}}} \right) - \left( {\dfrac{{x{y^2}}}{{x - y}}.\dfrac{{x - y}}{{ - {y^2}}}} \right)\)
\( = \dfrac{{{x^2}y}}{{{x^2}}} - \dfrac{{x{y^2}}}{{ - {y^2}}} = y + x = x + y\)
b) Với \(P = \dfrac{{2xy}}{{{x^2} - {y^2}}}\) và \(Q = \dfrac{{2xy}}{{{x^2} + {y^2}}}\)
Ta có:
\(\dfrac{{{P^2}{Q^2}}}{{{P^2} - {Q^2}}} = \dfrac{{{{\left( {\dfrac{{2xy}}{{{x^2} - {y^2}}}} \right)}^2}.{{\left( {\dfrac{{2xy}}{{{x^2} + {y^2}}}} \right)}^2}}}{{{{\left( {\dfrac{{2xy}}{{{x^2} - {y^2}}}} \right)}^2} - {{\left( {\dfrac{{2xy}}{{{x^2} + {y^2}}}} \right)}^2}}}\)
\( = \dfrac{{{{\left[ {\dfrac{{2xy.2xy}}{{\left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)}}} \right]}^2}}}{{\dfrac{{4{x^2}{y^2}}}{{{{\left( {{x^2} - {y^2}} \right)}^2}}} - \dfrac{{4{x^2}{y^2}}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}}}}\)
\( = \dfrac{{\dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}}{{\dfrac{{4{x^2}{y^2}{{\left( {{x^2} + {y^2}} \right)}^2} - 4{x^2}{y^2}{{\left( {{x^2} - {y^2}} \right)}^2}}}{{{{\left[ {\left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)} \right]}^2}}}}}\)
\( = \dfrac{{\dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}}{{\dfrac{{4{x^2}{y^2}\left[ {{{\left( {{x^2} + {y^2}} \right)}^2} - {{\left( {{x^2} - {y^2}} \right)}^2}} \right]}}{{{{\left[ {\left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)} \right]}^2}}}}}\)
\( = \dfrac{{\dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}}{{\dfrac{{4{x^2}{y^2}.({x^4} + 2{x^2}{y^2} + {y^4} - {x^4} + 2{x^2}{y^2} - {y^4})}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}}\)
\( = \dfrac{{\dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}}{{\dfrac{{4{x^2}{y^2}.4{x^2}{y^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}} = \dfrac{{\dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}}{{\dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}}}\)
\( = \dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}}:\dfrac{{{{\left( {4{x^2}{y^2}} \right)}^2}}}{{{{\left( {{x^4} - {y^4}} \right)}^2}}} = 1\)
