Question
Prove that : $\frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}=\frac{2}{b^2-a^2}$
✓
Answer
$ \frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}=\frac{2}{b^2-a^2}$
$ \text { L.H.S. }$
$=\frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}$
$ =\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{b+a}{a b}}+\frac{\frac{1}{a}}{\frac{b-a}{a b}}$
$ =\frac{1}{a} \times \frac{a b}{b+a}+\frac{1}{a} \times \frac{a b}{b-a}$
$ =\frac{b}{b+a}+\frac{b}{b-a}$
$ =\frac{b^2-a b+b^2+a b}{b^2-a^2}$
$ =\frac{2 b^2}{b^2-a^2}$
$ =\text { R.H.S. }$
$ \text { L.H.S. }$
$=\frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}$
$ =\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{b+a}{a b}}+\frac{\frac{1}{a}}{\frac{b-a}{a b}}$
$ =\frac{1}{a} \times \frac{a b}{b+a}+\frac{1}{a} \times \frac{a b}{b-a}$
$ =\frac{b}{b+a}+\frac{b}{b-a}$
$ =\frac{b^2-a b+b^2+a b}{b^2-a^2}$
$ =\frac{2 b^2}{b^2-a^2}$
$ =\text { R.H.S. }$
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