If `A+B=45^(@)`, find the valueof `tanA+tanB+tanA tanB`:

Options

A. `-1`
B. `(1)/(2)`
C. `sqrt3`
D. 1

Answer

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

### Step-by-Step Solution:

1. **Use the Angle Sum Identity for Tangent**:
We know that:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Since \( A + B = 45^\circ \), we have:
\[ \tan(45^\circ) = 1 \]
Therefore, we can write:
\[ 1 = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

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

3. **Rearranging the Equation**:
Rearranging the equation, we get:
\[ \tan A + \tan B + \tan A \tan B = 1 \]

4. **Conclusion**:
Hence, the value of \( \tan A + \tan B + \tan A \tan B \) is:
\[ \boxed{1} \]