Question

If \[ 2\tan^{-1}(\cos x) = \tan^{-1}(2 \csc x), \] then the value of \( x \) is:

• A. \(\frac{3\pi}{4}\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{\pi}{3}\)
• D. None of these

Hint:

Use double angle formulas for inverse tangent carefully.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Use the formula \(2\tan^{-1}y = \tan^{-1}\left(\frac{2y}{1-y^2}\right)\) for appropriate range.

Step 2: Detailed Explanation:
Let \(y = \cos x\). Then LHS = \(2\tan^{-1}y = \tan^{-1}\left(\frac{2y}{1-y^2}\right)\). Thus \(\tan^{-1}\left(\frac{2\cos x}{1-\cos^2 x}\right) = \tan^{-1}(2\cosec x)\). So \(\frac{2\cos x}{\sin^2 x} = 2\cosec x\). \[ \frac{2\cos x}{\sin^2 x} = \frac{2}{\sin x} \] \[ \cos x = \sin x \] \[ \tan x = 1 \] \[ x = \frac{\pi}{4} \] (in principal range).

Step 3: Final Answer:
\(x = \frac{\pi}{4}\), which corresponds to option (B).