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Question
if `tan^(-1) (x-1)/(x - 2) + tan^(-1) (x + 1)/(x + 2) = pi/4` then find the value of x.
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` `[tan^(-1) x + tan^(-1) y = tan^(-1) (x+y)/(1-xy)]`
`=> tan^(-1) [((x-1)(x+2)+(x+1)(x-2))/((x + 2)(x-2) - (x - 1)(x + 1)]] = pi/4`
`=> tan^(-1) [(x^2 + x - 2 + x^2 - x- 2)/(x^2 - 4 - x^2 + 1)] = pi/4`
`=> tan^(-1) [(2x^2 - 4)/(-3)] = pi/4`
`=> tan[tan^(-1) (4 - 2x^2)/3] = tan pi/4`
`=> (4- 2x^2)/3 = 1`
`=> 4 - 2x^2 = 3`
`=> 2x^2 = 4 - 3 =1`
`=> x = +- 1/sqrt2`
Hence, the value of x is `+- 1/sqrt2`
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