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Prove that (i) (a^(-1))/(a^(-1) + b^(...

Prove that
(i) `(a^(-1))/(a^(-1) + b^(-1)) + (a^(-1))/(a^(-1)-b^(-1)) = (2b^(2))/(b^(2) -a^(2))`
(ii) `(1)/(1+x^(a-b)) + (1)/(1+x^(b-a)) = 1`

Text Solution

Verified by Experts

We have .
(i) `(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).(ab)/((b+a)) +(1)/(a).(ab)/((b-a)) = (b)/(b+a) +(b)/(b-a)`
` = (b(b-a)+b(b+a))/((b+a)(b-a))=(b^(2) -ab+b^(2) + ab)/(b^(2) -a^(2))`
`= (2b^(2))/(b^(2) -a^(2))`
(ii) `(1)/(1+x^(b-a)) +(1)/(1+x^(b-a)) = (1)/(1+x^(a).x^(-b)) +(1)/(1+x^(a).x^(-b))+(1)/(1+x^(b) .x^(-a))`
` = (1)/(1+(x^(a))/(x^(b))) +(1)/(1+(x^(b))/(x^(a))) +(1)/(((x^(a) +x^(b))/x^(b))) +(1)/(((x^(a) + x^(b))/(x^(a))))`
` =(x^(b))/(x^(a) + x^(b)) +(x^(a))/(x^(a) + x^(b)) = (x^(b) + x^(b))/(x^(a) + x^(b)) = 1`
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