If A + B = 45°, then the value of 2(1 + tan A) (1 + tan B) is:

Options

A. 4
B. 1
C. 0
D. 2

Answer

To solve the problem, we need to find the value of \(2(1 + \tan A)(1 + \tan B)\) given that \(A + B = 45^\circ\).

### Step-by-step Solution:

1. **Use the identity for tangent**:
Since \(A + B = 45^\circ\), we can use the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Therefore, we have:
\[ \tan(45^\circ) = 1 \]
This gives us the equation:
\[ 1 = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

2. **Cross-multiply to simplify**:
Cross-multiplying gives:
\[ 1 - \tan A \tan B = \tan A + \tan B \]
Rearranging this, we get:
\[ 1 = \tan A + \tan B + \tan A \tan B \]

3. **Express \(1 + \tan A\) and \(1 + \tan B\)**:
We can express \(1 + \tan A\) and \(1 + \tan B\) as:
\[ 1 + \tan A = 1 + \tan A \]
\[ 1 + \tan B = 1 + \tan B \]

4. **Calculate \(2(1 + \tan A)(1 + \tan B)\)**:
Now, we can expand \(2(1 + \tan A)(1 + \tan B)\):
\[ 2(1 + \tan A)(1 + \tan B) = 2[(1 + \tan A) + (1 + \tan B) + \tan A \tan B] \]
Using the identity from step 2, we know that:
\[ \tan A + \tan B + \tan A \tan B = 1 \]
Thus:
\[ 2(1 + \tan A)(1 + \tan B) = 2(1 + 1) = 2 \times 2 = 4 \]

5. **Final Result**:
Therefore, the value of \(2(1 + \tan A)(1 + \tan B)\) is:
\[ \boxed{4} \]