If the sum of all the solutions of tan ⁻¹( 2x/(1 - x²) ) + cot ⁻¹( (1 - x²)/2x ) = π/3, - 1 < x < 1,x ≠ 0, is α - 4/√3, then α is equal to _____________.

Solution

Maths sir · 4 steps

Split by the sign of x: for negative x, the inverse cot term picks up an extra π.

1. First term. Since -1 < x < 1, the first term is tan⁻¹ 2x/(1-x²) = 2 tan⁻¹ x for every allowed x.
2. Second term. Use cot⁻¹ y = tan⁻¹ 1/y for y > 0 and cot⁻¹ y = π + tan⁻¹ 1/y for y < 0. Here y = (1-x²)/2x has the same sign as x. So the second term is 2 tan⁻¹ x when x > 0, and π + 2 tan⁻¹ x when x < 0.

   Case	Equation	Solution
   0 < x < 1	4 tan⁻¹ x = π/3, so tan⁻¹ x = π/12	x = tan 15° = 2 - √3
   -1 < x < 0	π + 4 tan⁻¹ x = π/3, so tan⁻¹ x = -π/6	x = -1/√3

3. Add them. Both values lie in (-1, 1), so both count. The sum is 2 - √3 - 1/√3 = 2 - (3 + 1)/√3 = 2 - 4/√3.
4. So. Comparing with α - 4/√3 gives α = 2.

The idea to keep. Converting cot⁻¹ y to tan⁻¹ 1/y works only for y > 0. For y < 0 you must add π, so always split the cases by sign before solving.

Answer from the NTA final answer key. Solution by PrepWiser.