`tan(tan^(-1)((1)/(2))-tan^(-1)((1)/(3)))=`

Options

A. `(1)/(6)`
B. `(1)/(7)`
C. `(5)/(6)`
D. `(7)/(6)`

Answer

To solve the question \( \tan(\tan^{-1}(\frac{1}{2}) - \tan^{-1}(\frac{1}{3})) \), we can use the formula for the tangent of the difference of two angles. The formula states:

\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

Here, we can let \( A = \tan^{-1}(\frac{1}{2}) \) and \( B = \tan^{-1}(\frac{1}{3}) \). Therefore, we have:

\[ \tan A = \frac{1}{2} \quad \text{and} \quad \tan B = \frac{1}{3} \]

Now, we can substitute these values into the formula:

\[ \tan(\tan^{-1}(\frac{1}{2}) - \tan^{-1}(\frac{1}{3})) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

Substituting the values of \( \tan A \) and \( \tan B \):

\[ = \frac{\frac{1}{2} - \frac{1}{3}}{1 + \left(\frac{1}{2}\right)\left(\frac{1}{3}\right)} \]

Next, we will simplify the numerator:

1. Find a common denominator for \( \frac{1}{2} \) and \( \frac{1}{3} \):
- The least common multiple of 2 and 3 is 6.
- Rewrite \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \).

Thus, the numerator becomes:

\[ \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

2. Now, simplify the denominator:

\[ 1 + \left(\frac{1}{2}\right)\left(\frac{1}{3}\right) = 1 + \frac{1}{6} = \frac{6}{6} + \frac{1}{6} = \frac{7}{6} \]

Now we can substitute back into our equation:

\[ \tan(\tan^{-1}(\frac{1}{2}) - \tan^{-1}(\frac{1}{3})) = \frac{\frac{1}{6}}{\frac{7}{6}} = \frac{1}{6} \cdot \frac{6}{7} = \frac{1}{7} \]

Thus, the final answer is:

\[ \tan(\tan^{-1}(\frac{1}{2}) - \tan^{-1}(\frac{1}{3})) = \frac{1}{7} \]