Question

prove that (question is given above in the picture)

Answer 1

Hope u like my process
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$\frac{ {a}^{ - 1} }{ {a }^{ - 1} + {b}^{ - 1} } + \frac{ {a}^{ - 1} }{ {a }^{ - 1} - {b}^{ - 1} } \\ = \frac{ {a}^{ - 1} ( {a}^{ - 1} - {b}^{ - 1}) + {a}^{ - 1 } ( {a}^{ - 1} + {b}^{ - 1} )}{( {a}^{ - 1} - {b}^{ - 1})( {a}^{ - 1} + {b}^{ - 1}) } \\ = \frac{ {a}^{ - 2} - {a}^{ - 1} {b}^{ - 1} + {a}^{ - 2} + {a}^{ - 1} {b}^{ - 1} }{ { ({a}^{ - 1}) }^{2} - { ({b}^{ - 1}) }^{2} } \\ = \frac{2 {a}^{ - 2} }{ {a}^{ - 2} - {b}^{ - 2} } \\ = \frac{2}{ {a}^{2} {a}^{ - 2} - \frac{ {a}^{2} }{ {b}^{2} } } = \frac{2 {b}^{2} }{ {b}^{2} - {a}^{2} } ......proved....$
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