Question

If \( \cos \theta = \frac{2 \cos \alpha + 1}{2 + \cos \alpha} \), then \( \tan^2 \left( \frac{\theta}{2} \right) \) is equal to:

• A. \( \frac{1}{3} \tan^2 \left( \frac{\alpha}{2} \right) \)
• B. \( \frac{1}{2} \tan^2 \left( \frac{\alpha}{2} \right) \)
• C. \( \frac{1}{3} \cos^2 \left( \frac{\alpha}{2} \right) \)
• D. \( \frac{1}{3} \cot^2 \left( \frac{\alpha}{2} \right) \)
• E. \( 3 \cot^2 \left( \frac{\alpha}{2} \right) \)

Hint:

Use trigonometric identities to simplify expressions for half-angles and other transformations.

Answer 1

The Correct Option is A

We are given the expression for \( \cos \theta \). Using the half-angle identity for \( \tan^2 \left( \frac{\theta}{2} \right) \), we apply the formula: \[ \tan^2 \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{1 + \cos \theta} \] Substituting \( \cos \theta = \frac{2 \cos \alpha + 1}{2 + \cos \alpha} \) into this identity, we simplify and find that: \[ \tan^2 \left( \frac{\theta}{2} \right) = \frac{1}{3} \tan^2 \left( \frac{\alpha}{2} \right) \] Thus, the correct answer is \( \frac{1}{3} \tan^2 \left( \frac{\alpha}{2} \right) \).