Solve : `tan^(-1)((x-1)/(x-2))+tan^(-1)((x+1)/(x+2))=(pi)/(4)`

Answer

`x = sqrt((5)/(2))`

Explanation

`tan^(-1).(x -1)/(x + 2) + tan^(-1).(x + 1)/(x + 2) = (pi)/(4)`
`rArr tan^(-1) [((x -1)/(x + 2) + (x + 1 )/(x + 2))/(1 - ((x -1)/(x + 2)) ((x +1)/(x + 2)))] = (pi)/(4)`
`rArr [(2x (x + 2))/(x^(2) + 4 + 4x -x^(2) + 1)] = tan.(pi)/(4)`
`rArr (2x (x + 2))/(4x + 5) = 1`
`rarr 2x^(2) + 4x = 4x + 5`
`:. x = +- sqrt((5)/(2))`
But for `x = -sqrt5//2`, L.H.S. is negative
Hence, `x = sqrt(5//2)`