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Question
Prove that: `a^-1/(a^-1+b^-1) + a^-1/(a^-1 - b^-1) = (2b^2)/(b^2 - a^2 )`
Solution
`a^-1/(a^-1+b^-1) + a^-1/(a^-1 - b^-1) = (2b^2)/(b^2 - a^2 )`
L.H.S. = `a^-1/(a^-1+b^-1) + a^-1/(a^-1 - b^-1)`
= `(1/a)/(1/a + 1/b) + (1/a)/(1/a - 1/b)`
= `(1/a)/((b + a)/(ab)) + (1/a)/((b - a)/(ab))`
= `1/a xx (ab)/(b+ a) + 1/a xx (ab)/(b - a)`
= `b/( b + a ) + b/(b - a)`
= `( b^2 - ab + b^2 + ab )/( b^2 - a^2 )`
= `( 2b^2 )/( b^2 - a^2 )`
= R.H.S.
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