Considering only the principal values of inverse trigonometric functions, the number of positive real values of x satisfying tan ⁻¹(x)+tan ⁻¹(2 x)=π/4 is :

Solution

Maths sir · 4 steps

Take tan of both sides to get a quadratic, then keep only the root that is positive.

1. Combine the inverses. For x > 0, the sum tan⁻¹ x + tan⁻¹ 2x equals π/4, which is less than π/2. That means x · 2x < 1, so the addition formula applies directly.

   (x + 2x)/(1 - 2x²) = tan π/4 = 1

2. Solve the quadratic. 3x = 1 - 2x² gives 2x² + 3x - 1 = 0. So x = (-3 ± √17)/4. The two roots are about 0.28 and -1.78.
3. Check the root. Only x = (√17 - 3)/4 ≈ 0.28 is positive. Here 2x² ≈ 0.16 < 1, so this root satisfies the condition from the first step.
4. So. There is exactly one positive solution. You can confirm this another way: for x > 0 the left side increases steadily from 0 towards π, so it passes π/4 only once. The answer is (d).

The idea to keep. After using the tan⁻¹ addition formula, check each root against the domain asked for. A sum of increasing inverse functions can hit a given value only once.

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