Question

If \( \tan \left( \frac{\pi}{12} + 2x \right) = \cot 3x \), where \( 0<x<\frac{\pi}{2} \), then the value of \( x \) is:

• A. \( \frac{\pi}{12} \)
• B. \( 3 \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)
• E. \( \frac{\pi}{24} \)

Hint:

Use trigonometric identities like \( \cot \theta = \frac{1}{\tan \theta} \) to simplify equations involving trigonometric functions.

Answer 1

The Correct Option is A

We are given the equation \( \tan \left( \frac{\pi}{12} + 2x \right) = \cot 3x \). 
Using the identity \( \cot \theta = \frac{1}{\tan \theta} \), we get: \[ \tan \left( \frac{\pi}{12} + 2x \right) = \frac{1}{\tan 3x} \] Multiplying both sides by \( \tan 3x \), we get: \[ \tan \left( \frac{\pi}{12} + 2x \right) \tan 3x = 1 \] Now solve for \( x \) by substituting values: \[ x = \frac{\pi}{12} \]