If x=2ab/b^2+1 Find the value of …

Given: $x = {{2ab} \over {{b^2} + 1}}$

${{\sqrt {a + x} + \sqrt {a - x} } \over {\sqrt {a + x} - \sqrt {a - x} }}$

= ${{\sqrt {a + x} + \sqrt {a - x} } \over {\sqrt {a + x} - \sqrt {a - x} }} \times {{\sqrt {a + x} + \sqrt {a - x} } \over {\sqrt {a + x} + \sqrt {a - x} }}$

= ${{{{\left( {\sqrt {a + x} } \right)}^2} + {{\left( {\sqrt {a - x} } \right)}^2} + 2\sqrt {a + x} .\sqrt {a - x} } \over {{{\left( {\sqrt {a + x} } \right)}^2} - {{\left( {\sqrt {a - x} } \right)}^2}}}$

= ${{a + x + a - x + 2\sqrt {{a^2} - {x^2}} } \over {a + x - a + x}}$

= ${{2a + 2\sqrt {{a^2} - {x^2}} } \over {2x}}$

= ${{a + \sqrt {{a^2} - {x^2}} } \over x}$

= ${{a + \sqrt {{a^2} - {{\left( {{{2ab} \over {{b^2} + 1}}} \right)}^2}} } \over {{{2ab} \over {{b^2} + 1}}}}$

= $a + \sqrt {{a^2} - {{4{a^2}{b^2}} \over {{{\left( {{b^2} + 1} \right)}^2}}}} \times {{{b^2} + 1} \over {2ab}}$

= $a + a\sqrt {{{{{\left( {{b^2} + 1} \right)}^2} - 4{b^2}} \over {{{\left( {{b^2} + 1} \right)}^2}}}} \times {{{b^2} + 1} \over {2ab}}$

= $a\left[ {1 + {1 \over {{b^2} + 1}}\sqrt {{b^4} + 1 + 2{b^2} - 4{b^2}} } \right] \times {{{b^2} + 1} \over {2ab}}$

= ${{{b^2} + 1 + \sqrt {{b^4} + 1 - 2{b^2}} } \over {{b^2} + 1}} \times {{{b^2} + 1} \over {2b}}$

= ${{{b^2} + 1 + \sqrt {{{\left( {{b^2} - 1} \right)}^2}} } \over {2b}}$

= ${{{b^2} + 1 + {b^2} - 1} \over {2b}}$

= ${{2{b^2}} \over {2b}}$

= $b$