Prove that $ \frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}=\frac{2 b^{2}}{b^{2}-a^{2}} $

Given: $( \frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}=\frac{2 b^{2}}{b^{2}-a^{2}})$

To do: To prove that $L.H.S.=R.H.S.$ 

Solution:  Given Expression:$\frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}$

$L.H.S.=\frac{\frac{1}{a}}{\frac{1}{a}+\frac{1}{b}}+\frac{\frac{1}{a}}{\frac{1}{a}-\frac{1}{b}}$

$=\frac{\frac{1}{a}}{\frac{a+b}{ab}}+\frac{\frac{1}{a}}{\frac{b-a}{ab}}$

$=\frac{ab}{a( a+b)}+\frac{ab}{a( b-a)}$

$=\frac{b}{b+a}+\frac{b}{b-a}$

$=\frac{b( b-a)+b( b+a)}{( b+a)( b-a)}$

$=\frac{b^{2}-ab+b^{2}+ab}{b^{2}-a^{2}}$

$=\frac{2b^{2}}{b^{2}-a^{2}}$

$=R.H.S.$

Hence, Proved that $( \frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}=\frac{2 b^{2}}{b^{2}-a^{2}})$

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Updated on: 10-Oct-2022

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