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

Options

A. 1
B. 2
C. -2
D. Not defined

Explanation

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. **Start with the expression**:
\[ (1 + \tan \alpha)(1 + \tan \beta) \]

2. **Expand the expression**:
\[ = 1 + \tan \alpha + \tan \beta + \tan \alpha \tan \beta \]

3. **Use the identity for \(\tan(\alpha + \beta)\)**:
We know that:
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]
Since \( \alpha + \beta = \frac{\pi}{4} \), we have:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Therefore:
\[ 1 = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]

4. **Cross-multiply to simplify**:
\[ 1 - \tan \alpha \tan \beta = \tan \alpha + \tan \beta \]

5. **Rearranging gives us**:
\[ \tan \alpha + \tan \beta + \tan \alpha \tan \beta = 1 \]

6. **Substituting back into the expanded expression**:
From step 2, we have:
\[ 1 + \tan \alpha + \tan \beta + \tan \alpha \tan \beta = 1 + 1 = 2 \]

7. **Final result**:
Thus, we conclude that:
\[ (1 + \tan \alpha)(1 + \tan \beta) = 2 \]

### Answer:
\[ \text{The value of } (1 + \tan \alpha)(1 + \tan \beta) \text{ is } 2. \]

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