If f(\theta)=\frac{1}{1+\tan \theta} and \alpha+\beta=\frac{5\pi}{4}, then what is the value of f(\alpha)f(\beta)?
- A. -\frac{1}{2}
- B. \frac{1}{2} ✓
- C. 1
- D. 2
Correct Answer: B. \frac{1}{2}
Explanation
Evaluate f(\alpha)f(\beta) = \frac{1}{(1+\tan \alpha)(1+\tan \beta)}. Given \alpha+\beta = \frac{5\pi}{4}, taking \tan on both sides gives \frac{\tan \alpha+\tan \beta}{1-\tan \alpha\tan \beta} = 1. Rearranging yields \tan \alpha+\tan \beta = 1-\tan \alpha\tan \beta, which implies 1 + \tan \alpha + \tan \beta + \tan \alpha \tan \beta = 2, or (1+\tan \alpha)(1+\tan \beta) = 2. Therefore, the product is \frac{1}{2}.