If the sum of all the solutions of tan - 1 2 x 1 - x 2 + cot - 1 1 - x 2 2 x = π 3 , - 1 < x < 1 , x ≠ 0 , is α - 4 3 , then α is equal to _____ .

Step-by-step solution

Given, tan - 1 2 x 1 - x 2 + cot - 1 1 - x 2 2 x = π 3 where x ∈ - 1 , 1 Case 1 : When x < 0 , then tan - 1 2 x 1 - x 2 + cot - 1 1 - x 2 2 x = π 3 ⇒ 2 tan - 1 x + π + 2 tan - 1 x = π 3 ⇒ 4 tan - 1 x = - π + π 3 ⇒ 4 tan - 1 x = - 2 π 3 ⇒ tan - 1 x = - π 6 ⇒ x = - 1 3 Case 2 : When x ≥ 0 , tan - 1 2 x 1 - x 2 + cot - 1 1 - x 2 2 x = π 3 ⇒ 2 tan - 1 x + 2 tan - 1 x = π 3 ⇒ tan - 1 x = π 12 ⇒ x = 2 -