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Question
If A = `x/(x + 1)` B = `1/(x + 1)` prove that `(("A" + "B")^2 + ("A" - "B")^2)/("A" + "B") = (2(x^2 + 1))/(x(x + 1)^2`
Solution
(A + B)2 = `(x/(x + 1) + 1/(x + 1))^2`
= `((x + 1)/(x + 1))^2`
= 1
(A – B)2 = `(x/(x + 1) - 1/(x + 1))^2`
= `((x - 1)/(x + 1))^2`
= `(x - 1)^2/(x + 1)^2`
`"A"/"B" = x/(x + 1) ÷ 1/(x + 1)`
= `x/(x + 1) xx (x + 1)/1`
= x
L.H.S. = `(("A" + "B")^2 + ("A" - "B")^2)/("A"/"B")`
= `1 + (x - 1)^2/(x + 1)^2 ÷ x`
= `((x + 1)^2 + (x - 1)^2)/(x + 1)^2 xx 1/x`
= `(x^2 + 2x + 1 + x^2 - 2x + 1)/(x(x + 1)^2`
= `(2x^2 + 2)/(x(x + 1)^2`
= `(2(x^2 + 1))/(x(x + 1)^2`
L.H.S. = R.H.S.
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