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Question
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
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Solution
`tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`
∴ `tan^-1 [("x - 1"/"x - 2" + "x + 1"/"x + 2")/(1 - ("x - 1"/"x - 2")("x + 1"/"x + 2"))] = pi/4`
∴ `(("x - 1")("x + 2") + ("x + 1")("x - 2"))/(("x - 2")("x + 2") - ("x - 1")("x + 1")) = tan pi/4`
∴ `(("x"^2 + "x" - 2) + ("x"^2 - "x" - 2))/(("x"^2 - 4) - ("x"^2 - 1)) = 1`
∴ `("x"^2 + "x" - 2 + "x"^2 - "x" - 2)/("x"^2 - 4 - "x"^2 + 1) = 1`
∴ `(2"x"^2 - 4)/-3 = 1`
∴ 2x2 - 4 = - 3
∴ 2x2 = 1
∴ x2 = `1/2`
∴ x = `+- 1/sqrt2`.
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