If `tanA=(1)/(2)andtanB=(1)/(3)`, then what is the value of (A + B) ?

Options

A. 0
B. `(pi)/(4)`
C. `(pi)/(2)`
D. `pi`

Explanation

To find the value of \( A + B \) given that \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \), we can use the formula for the tangent of the sum of two angles:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

### Step-by-step solution:

1. **Substitute the values of \( \tan A \) and \( \tan B \)**:
\[ \tan(A + B) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right)} \]

2. **Calculate the numerator**:
- Find a common denominator for \( \frac{1}{2} \) and \( \frac{1}{3} \):
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

3. **Calculate the denominator**:
- Calculate \( \tan A \tan B \):
\[ \tan A \tan B = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]
- Substitute this into the denominator:
\[ 1 - \tan A \tan B = 1 - \frac{1}{6} = \frac{5}{6} \]

4. **Combine the results**:
\[ \tan(A + B) = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \]

5. **Find the angle \( A + B \)**:
- Since \( \tan(A + B) = 1 \), we know that:
\[ A + B = \frac{\pi}{4} \quad \text{(or } 45^\circ\text{)} \]

### Final Answer:
\[ A + B = \frac{\pi}{4} \]