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

The Correct Option is B

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