If `tantheta=(1)/(2) and tanphi=(1)/(3)`, then the value of `theta+phi` is

Options

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

Explanation

To solve the problem where \( \tan \theta = \frac{1}{2} \) and \( \tan \phi = \frac{1}{3} \), we need to find the value of \( \theta + \phi \).

### Step-by-step Solution:

1. **Use the formula for the tangent of a sum**:
The formula for the tangent of the sum of two angles is given by:
\[ \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]

2. **Substitute the values of \( \tan \theta \) and \( \tan \phi \)**:
We know that \( \tan \theta = \frac{1}{2} \) and \( \tan \phi = \frac{1}{3} \). Substituting these values into the formula:
\[ \tan(\theta + \phi) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right)} \]

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.
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
Therefore:
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

4. **Calculate the denominator**:
Now, calculate \( 1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right) \):
\[ \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]
Thus:
\[ 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

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

6. **Determine \( \theta + \phi \)**:
Since \( \tan(\theta + \phi) = 1 \), we know that:
\[ \theta + \phi = \frac{\pi}{4} \]

### Final Answer:
Thus, the value of \( \theta + \phi \) is \( \frac{\pi}{4} \).

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