If `A + B = 45^(@)`, then find the value of `(1+tan A)(1+ tan B)`

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To solve the problem where \( A + B = 45^\circ \) and we need to find the value of \( (1 + \tan A)(1 + \tan B) \), we can follow these steps:

### Step-by-Step Solution:

1. **Use the Sum of Angles Formula for Tangent:**
Since \( A + B = 45^\circ \), we can apply the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Given \( A + B = 45^\circ \), we know that:
\[ \tan(45^\circ) = 1 \]
Therefore, we can set up the equation:
\[ 1 = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

2. **Cross Multiply:**
Cross multiplying gives us:
\[ 1 - \tan A \tan B = \tan A + \tan B \]

3. **Rearranging the Equation:**
Rearranging the equation leads to:
\[ \tan A + \tan B + \tan A \tan B = 1 \]

4. **Adding 1 to Both Sides:**
Now, we add 1 to both sides of the equation:
\[ 1 + \tan A + \tan B + \tan A \tan B = 2 \]

5. **Factor the Left Side:**
The left side can be factored as:
\[ (1 + \tan A)(1 + \tan B) = 2 \]

6. **Conclusion:**
Thus, we find that:
\[ (1 + \tan A)(1 + \tan B) = 2 \]

### Final Answer:
The value of \( (1 + \tan A)(1 + \tan B) \) is \( 2 \).