If \( \cos\alpha + \cos\beta = 0 = \sin\alpha + \sin\beta \), \( \alpha \neq \beta \) then what is a value of \( \cos 2\alpha + \cos 2\beta + 2\cos(\alpha+\beta) \) ?
- A. 0 ✓
- B. 1
- C. 2
- D. 4
Correct Answer: A. 0
Explanation
The given expression can be written as \cos^2\alpha - \sin^2\alpha + \cos^2\beta - \sin^2\beta + 2(\cos\alpha\cos\beta - \sin\alpha\sin\beta), which simplifies exactly to (\cos\alpha + \cos\beta)^2 - (\sin\alpha + \sin\beta)^2. Since both sums are zero, the expression evaluates to 0 - 0 = 0.