If `tan A=1/2` and `tan B=1/3`, then: A+B=

Options

A. `pi/4`
B. 0
C. `pi`
D. `pi/6`

Answer

To solve the problem, we need to find the value of \( A + B \) given that \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \).

### Step-by-Step Solution:

1. **Use the formula for \( \tan(A + B) \)**:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

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

3. **Calculate the numerator**:
- Find a common denominator for \( \frac{1}{2} \) and \( \frac{1}{3} \):
\[ \text{LCM of 2 and 3 is 6} \]
Therefore,
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
Thus,
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

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

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

6. **Find \( A + B \)**:
- Since \( \tan(A + B) = 1 \), we take the inverse tangent:
\[ A + B = \tan^{-1}(1) \]
- We know that \( \tan^{-1}(1) = \frac{\pi}{4} \) (or 45 degrees).

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