Question

If \( \cos x = \frac{2\cos y - 1}{2 - \cos y} \), where \( x, y \in (0, \pi) \), then \( \tan \frac{x}{2} \cot \frac{y}{2} \) is equal to:

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $1/\sqrt{2}$
• D. $1/\sqrt{3}$

Hint:

If $\cos x = 2 \cos y - 12 - \cos y}$, where $x, y \in (0, π)$, then $\tan \fracx/2 \cot y/2$ is equal to

Answer 1

The Correct Option is B

Step 1: Concept
Use the half-angle formula: $\cos \theta = \frac{1 - \tan^2(\theta/2)}{1 + \tan^2(\theta/2)}$.
Step 2: Analysis
Substitute the formula for both $\cos x$ and $\cos y$ into the given equation.
Step 3: Evaluation
Applying Componendo and Dividendo or simplifying the resulting algebraic expression leads to $6 \tan^2(y/2) = 2 \tan^2(x/2)$.
Step 4: Conclusion
This simplifies to $\frac{\tan^2(x/2)}{\tan^2(y/2)} = 3$, so $\tan(x/2) \cot(y/2) = \sqrt{3}$.
Final Answer: (b)