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Question
If A = `(2x + 1)/(2x - 1)`, B = `(2x - 1)/(2x + 1)` find `1/("A" - "B") - (2"B")/("A"^2 - "B"^2)`
Solution
`1/("A" - "B") - (2"B")/("A"^2 - "B"^2) = 1/("A" - "B") - (2"B")/(("A" + "B")("A" - "B"))`
= `("A" + "B" - 2"B")/(("A" + "B")("A" - "B"))`
= `("A" - "B")/(("A" + "B")("A" - "B"))`
= `1/("A" + "B")`
= `1 ÷ (2x + 1)/(2x - 1) + (2x - 1)/(2x + 1)`
= `1 ÷ ((2x + 1)^2 + (2x - 1)^2)/((2x - 1)(2x + 1))`
= `1 ÷ (4x^2 + 1 + 4x + 4x^2 + 1 - 4x)/((2x - 1)(2x + 1))`
= `1 ÷ (8x^2 + 2)/((2x - 1)(2x + 1))`
= `1 ÷ (2(4x^2 + 1))/(4x^2 - 1)`
= `1 xx (4x^2 - 1)/(2(4x^2 + 1))`
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