Question

If \[ \cos x + \cos y = -\cos \alpha, \quad \sin x + \sin y = -\sin \alpha, \quad \text{then} \quad \cot \left( \frac{x + y}{2} \right) = \]

• A. \( -\cot \alpha \)
• B. \( \cot \alpha \)
• C. \( -\tan \alpha \)
• D. \( \tan \alpha \)

Hint:

For equations involving trigonometric functions, use sum and difference identities to simplify and find relationships between angles.

Answer 1

The Correct Option is B

Step 1: Use the sum of angles identity.
We are given the equations for \( \cos x + \cos y \) and \( \sin x + \sin y \). By using the sum of angles identity for sine and cosine, we find that: \[ \cot \left( \frac{x + y}{2} \right) = \cot \alpha \]
Step 2: Conclusion.
Thus, \( \cot \left( \frac{x + y}{2} \right) = \cot \alpha \), corresponding to option (B).