Question

Solve: tan^-1 4 x + tan^-1 6x `= π/(4)`.

Answer 1

We have `tan^-1 4 "x" + tan^-1 6"x" = π/(4)`

⇒ `tan ( tan^-1 4"x" + tan^-1 6"x") = tan  π/(4)`

⇒ `(tan(tan^-1 4"x") + tan(tan^-1 6"x"))/(1-tan(tan^-1 4"x")·tan(tan^-1 6"x")` = 1

⇒ `(4"x" + 6"x")/(1-4"x"  6"x")` = 1

⇒ `(10"x")/(1 - 24"x"^2)` = 1

⇒ 24x² + 10x - 1 = 0

⇒ 24x² + 12x - 2x - 1 = 0

⇒ 12x (2x + 1) - 1 (2x + 1) = 0

⇒ (2x + 1) (12x - 1) = 0

⇒ `"x" = -(1)/(2) , (1)/(12)`

But `"x" = (-1)/(2)`  does not satisfy the equation as the LHS will become negative. Therefore, the value of `"x"  "is" (1)/(12)`.