Question

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan^{-1}(x) + \tan^{-1}(2x) = \frac{\pi}{4}$ is:

• A. More than 2
• B. 1
• C. 2
• D. 0

Answer 1

Answer: (B)

1

Answer 2

Given: $\tan^{-1} x + \tan^{-1} 2x = \frac{\pi}{4}$, where $x > 0$.

$\implies \tan^{-1} 2x = \frac{\pi}{4} - \tan^{-1} x$

Taking tangent on both sides:

$\implies 2x = \frac{1 - x}{1 + x}$

$\implies 2x(1 + x) = 1 - x$

$\implies 2x^2 + 3x - 1 = 0$

Solving the quadratic equation:

$x = \frac{-3 \pm \sqrt{9 + 8}}{4}$

$x = \frac{-3 \pm \sqrt{17}}{4}$

Since $x > 0$, the only possible solution is:

$x = \frac{-3 + \sqrt{17}}{4}$

Thus, the number of positive real values of $x$ is $1$.

The Correct answer is: 1