Question

Solve for \( x \), \[ 2 \tan^{-1} x + \sin^{-1} \left( \frac{2x}{1 + x^2} \right) = 4\sqrt{3} \]

• A. \( x = \sqrt{3} \)
• B. \( x = 1 \)
• C. \( x = 2 \)
• D. \( x = 0 \)

Hint:

Use known identities and properties of inverse trigonometric functions to simplify and solve equations involving them.

Answer 1

The Correct Option is B

To solve the equation, we start by simplifying each term: - The first term is \( 2 \tan^{-1} x \). We will consider the inverse tangent function and express \( \tan^{-1} x \) in terms of an angle \( \theta \), where \( \tan \theta = x \). - The second term involves \( \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), which is a standard identity for \( \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), which equals \( \tan^{-1} x \). Thus, the equation becomes: \[ 2 \tan^{-1} x + \tan^{-1} x = 4\sqrt{3} \] Simplifying further: \[ 3 \tan^{-1} x = 4\sqrt{3} \] \[ \tan^{-1} x = \frac{4\sqrt{3}}{3} \] Now, take the tangent of both sides: \[ x = \tan \left( \frac{4\sqrt{3}}{3} \right) \] The solution gives \( x = 1 \).