If tan `A=1//2andtanB=1//3`, then what is the value of 4A +4B ?

Options

A. `pi//4`
B. `pi//2`
C. `pi`
D. `2pi`

Explanation

To solve the problem, we need to find the value of \(4A + 4B\) given that \(\tan A = \frac{1}{2}\) and \(\tan B = \frac{1}{3}\).

### Step-by-Step Solution:

1. **Use the tangent addition formula**:
The formula for the tangent of the sum of two angles is:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Here, we have \(\tan A = \frac{1}{2}\) and \(\tan B = \frac{1}{3}\).

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

3. **Calculate the numerator**:
To add \(\frac{1}{2}\) and \(\frac{1}{3}\), we need a common denominator. The least common multiple of 2 and 3 is 6.
\[ \tan A + \tan B = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

4. **Calculate the denominator**:
First, 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:
\[ 1 - \tan A \tan B = 1 - \frac{1}{6} = \frac{5}{6} \]

5. **Combine the results**:
Now substitute the values back into the formula:
\[ \tan(A + B) = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \]

6. **Find \(A + B\)**:
Since \(\tan(A + B) = 1\), we know that:
\[ A + B = \frac{\pi}{4} + n\pi \quad \text{for some integer } n \]
For the principal value, we take \(n = 0\):
\[ A + B = \frac{\pi}{4} \]

7. **Calculate \(4A + 4B\)**:
We can express \(4A + 4B\) as:
\[ 4A + 4B = 4(A + B) = 4 \cdot \frac{\pi}{4} = \pi \]

### Final Answer:
Thus, the value of \(4A + 4B\) is \(\pi\).