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 trigonometric identities and formulas like the double angle formula for tangent and the identity \( \csc x = \frac{1}{\sin x} \) to simplify and solve equations involving inverse trigonometric functions.

Answer 1

The Correct Option is B

Step 1: Understanding the given equation.
We are given the equation: \[ 2 \tan^{-1} (\cos x) = \tan^{-1} (2 \csc x). \] To simplify this, let us use the formula for the tangent of a double angle: \[ \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}. \] We can apply this formula to the left-hand side of the equation. Let: \[ \theta = \tan^{-1} (\cos x), \] so that: \[ \tan \theta = \cos x. \] Using the double angle formula for tangent: \[ \tan(2\theta) = \frac{2 \cos x}{1 - \cos^2 x}. \] Thus, the equation becomes: \[ \frac{2 \cos x}{1 - \cos^2 x} = 2 \csc x. \]
Step 2: Simplifying the equation.
Now, use the identity \( \csc x = \frac{1}{\sin x} \) to rewrite the equation: \[ \frac{2 \cos x}{1 - \cos^2 x} = \frac{2}{\sin x}. \] Simplifying further: \[ \frac{\cos x}{1 - \cos^2 x} = \frac{1}{\sin x}. \] Now, solve for \( x \) by recognizing that this equation is satisfied when \( x = \frac{\pi}{4} \).
Step 3: Conclusion.
Therefore, the value of \( x \) is \( \frac{\pi}{4} \), and the correct answer is (b).