Question

The number of values of \( x \) satisfying \( \tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6} \) and \( x<\left[ \frac{-1}{2\sqrt{6}} , \frac{1}{2\sqrt{6}} \right] \) is:

• A. 1
• B. 0
• C. 2
• D. 3

Hint:

When solving trigonometric equations involving arctangents, use the addition formula and carefully solve the resulting equation.

Answer 1

The Correct Option is A

Step 1: Use the addition formula for arctangents.
We use the formula for the sum of two arctangents: \[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left( \frac{a + b}{1 - ab} \right) \] So the equation becomes: \[ \tan^{-1}(4x) + \tan^{-1}(6x) = \tan^{-1}\left( \frac{4x + 6x}{1 - (4x)(6x)} \right) = \frac{\pi}{6} \] Step 2: Solve for \( x \).
We have: \[ \tan^{-1}\left( \frac{10x}{1 - 24x^2} \right) = \frac{\pi}{6} \] This implies: \[ \frac{10x}{1 - 24x^2} = \tan\left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}} \] Step 3: Solve the resulting equation.
Now, solve for \( x \). This will yield only one solution that satisfies the given condition. Step 4: Conclusion.
Thus, the number of values of \( x \) that satisfy the equation is 1. Final Answer: \[ \boxed{1} \]