If 2tanA = 3tanB = 1, then what is tan(A-B) equal to?

Options

A. `1//5`
B. `1//6`
C. `1//7`
D. `1//9`

Answer

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

### Step-by-Step Solution:

1. **Express \( \tan A \) and \( \tan B \) in terms of a common variable:**
Since \( 2\tan A = 1 \), we can express \( \tan A \) as:
\[ \tan A = \frac{1}{2} \]
Similarly, since \( 3\tan B = 1 \), we can express \( \tan B \) as:
\[ \tan B = \frac{1}{3} \]

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

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

4. **Calculate the numerator:**
To calculate the numerator \( \frac{1}{2} - \frac{1}{3} \), we need a common denominator:
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
Thus,
\[ \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

5. **Calculate the denominator:**
Now calculate the denominator \( 1 + \left(\frac{1}{2} \cdot \frac{1}{3}\right) \):
\[ \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]
Therefore,
\[ 1 + \frac{1}{6} = \frac{6}{6} + \frac{1}{6} = \frac{7}{6} \]

6. **Combine the results:**
Now we can combine the results:
\[ \tan(A - B) = \frac{\frac{1}{6}}{\frac{7}{6}} = \frac{1}{6} \cdot \frac{6}{7} = \frac{1}{7} \]

### Final Answer:
Thus, the value of \( \tan(A - B) \) is:
\[ \tan(A - B) = \frac{1}{7} \]