Question

Solve: tan^-1 (x + 1) + tan^-1 (x – 1) = `tan^-1 (4/7)`

Answer 1

tan^-1 (x + 1) + tan^-1 (x – 1) = `tan^-1 (4/7)`

`tan^-1 (((x + 1) + (x - 1))/(1 - (x + 1)(x - 1))) = tan^-1 (4/7)`

`tan^-1 ((2x)/(1 - (x^2 - 1))) = tan^-1 (4/7)`

`tan^-1 ((2x)/(1 - x^2 + 1)) = tan^-1 (4/7)`

`tan^-1 ((2x)/(2 - x^2)) = tan^-1 (4/7)`

∴ `(2x)/(2 - x^2) = 4/7`

`(x)/(2 - x^2) = 2/7`

⇒ 7x = 2(2 – x²)

⇒ 7x = 4 – 2x²

⇒ 2x² + 7x – 4 = 0

⇒ (x + 4) (2x – 1) = 0

⇒ x + 4 = 0 (or) 2x – 1 = 0

⇒ x = -4 (or) x = `1/2`

x = -4 is rejected, since does not satisfies the question.

∴ x = `1/2`