If `tan A=1/2` and `tan B=1/3`, then `tan(A +B)` is equal to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find \( \tan(A + B) \) 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. **Identify the values of \( \tan A \) and \( \tan B \)**:
\[ \tan A = \frac{1}{2}, \quad \tan B = \frac{1}{3} \]

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

3. **Calculate \( \tan A + \tan B \)**:
\[ \tan A + \tan B = \frac{1}{2} + \frac{1}{3} \]
To add these fractions, find a common denominator (which is 6):
\[ \tan A + \tan B = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

4. **Calculate \( \tan A \tan B \)**:
\[ \tan A \tan B = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \]

5. **Substitute back into the formula**:
\[ \tan(A + B) = \frac{\frac{5}{6}}{1 - \frac{1}{6}} \]

6. **Calculate the denominator**:
\[ 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

7. **Now substitute the denominator back into the equation**:
\[ \tan(A + B) = \frac{\frac{5}{6}}{\frac{5}{6}} \]

8. **Simplify the expression**:
\[ \tan(A + B) = 1 \]

### Final Answer:
\[ \tan(A + B) = 1 \]