Solve : ` tan ^(-1) 2x + tan ^(-1) 3x = pi/4`

Explanation

While solving such equations we use the following formulas:
`tan^(1) x + tan^(-1) y = tan^(-1).(x + y)/(1 - xy)`...(i)
We know that this formula is valid only when `xy lt 1`.
So, we may get some extra solutions by solving equation with this formula. The extra solutions can be removed from the solution set by putting the values obtained in the original equation and checking whether it satisfies the equation or not
`tan^(-1) 2x + tan^(-1) 3x = (pi)/(4)`
or `tan^(-1) ((2x + 3x)/(1 - 6x^(2))) = (pi)/(4)`
or `(5x)/(1 = 6x^(2)) = 1`
or `6x^(2) + 5x -1 = 0`
or `(6x -1) (x + 1) = 0`
or `x = 1//6, -1`
But for `x = -1, tan^(-1) 2x + tan^(-1) 3x lt 0`
So, it does not satisfy Eq. (i)
Hence, `x = (1)/(6)`