If `alpha+beta=pi/(4), then (1+"tan"alpha)(1+"tan"beta)` is

Options

A. 1
B. 2
C. `-1`
D. `-2`

Answer

To solve the problem, we need to find the value of \( (1 + \tan \alpha)(1 + \tan \beta) \) given that \( \alpha + \beta = \frac{\pi}{4} \).

### Step-by-Step Solution:

1. **Use the given condition**: We know that \( \alpha + \beta = \frac{\pi}{4} \). We can express \( \alpha \) in terms of \( \beta \):
\[ \alpha = \frac{\pi}{4} - \beta \]

2. **Substitute into the expression**: We need to find \( (1 + \tan \alpha)(1 + \tan \beta) \). Substitute \( \alpha \):
\[ (1 + \tan(\frac{\pi}{4} - \beta))(1 + \tan \beta) \]

3. **Use the tangent subtraction formula**: The formula for \( \tan(\frac{\pi}{4} - \beta) \) is:
\[ \tan(\frac{\pi}{4} - \beta) = \frac{1 - \tan \beta}{1 + \tan \beta} \]
Therefore, we can rewrite the expression:
\[ (1 + \frac{1 - \tan \beta}{1 + \tan \beta})(1 + \tan \beta) \]

4. **Simplify the first part**: Now simplify \( 1 + \frac{1 - \tan \beta}{1 + \tan \beta} \):
\[ = \frac{(1 + \tan \beta) + (1 - \tan \beta)}{1 + \tan \beta} \]
\[ = \frac{2}{1 + \tan \beta} \]

5. **Combine the expressions**: Now substitute this back into the expression:
\[ \frac{2}{1 + \tan \beta}(1 + \tan \beta) \]

6. **Final simplification**: The \( (1 + \tan \beta) \) terms cancel out:
\[ = 2 \]

### Conclusion:
Thus, the value of \( (1 + \tan \alpha)(1 + \tan \beta) \) is \( 2 \).