Cho \(P = \left( {\frac{{\sqrt x + \sqrt y }}{{1 - \sqrt {xy} }} + \frac{{\sqrt x - \sqrt y }}{{1 + \sqrt {xy} }}} \right):\left( {1 + \frac{{x + y + 2xy}}{{1 - xy}}} \right)\).

a. Rút gọn P.

b. Tính giá trị của P khi \(x = \frac{2}{{2 + \sqrt 3 }}\) .

ĐK: \(xy \ne 1,xy \ge 0\)

a. \(P = \frac{{\left( {\sqrt x + \sqrt y } \right)\left( {1 + \sqrt {xy} } \right) + \left( {\sqrt x + \sqrt y } \right)\left( {1 - \sqrt {xy} } \right)}}{{1 - xy}}:\frac{{1 - xy + x + y + 2xy}}{{1 - xy}}\) \(\)

\(P = \frac{{\sqrt x + \sqrt y + x\sqrt y + y\sqrt x + \sqrt x - x\sqrt y - \sqrt y + y\sqrt x }}{{1 - xy}}:\frac{{1 + x + y + xy}}{{1 - xy}}\)

\(P = \frac{{2\sqrt x + 2y\sqrt x }}{{1 - xy}}.\frac{{1 - xy}}{{\left( {1 + x} \right)\left( {1 + y} \right)}} = \frac{{2\sqrt x \left( {1 + y} \right)}}{{\left( {1 + x} \right)\left( {1 + y} \right)}} = \frac{{2\sqrt x }}{{1 + x}}\)

b. Khi \(x = \frac{2}{{2 + \sqrt 3 }} = \frac{{2\left( {2 - \sqrt 3 } \right)}}{{{2^2} - \sqrt {{3^2}} }} = \frac{{4 - 2\sqrt 3 }}{1}\) hay \(x = {\left( {\sqrt 3 - 1} \right)^2}\)

Khi đó \(P = \frac{{2{{\sqrt {\left( {\sqrt 3 - 1} \right)} }^2}}}{{1 + 4 - 2\sqrt 3 }} = \frac{{2\left( {\sqrt 3 - 1} \right)}}{{5 - 2\sqrt 3 }}\)

\(P = \frac{{\left( {2\sqrt 3 - 2} \right)\left( {5 + 2\sqrt 3 } \right)}}{{{5^2} - {{\left( {2\sqrt 3 } \right)}^2}}} = \frac{{2\left( {3\sqrt 3 + 1} \right)}}{{13}}\)