`4 tan^(-1)(1/5)-tan^(-1)(1/239)` is equal o

Options

A. `pi`
B. `pi//2`
C. `pi//3`
D. `pi//4`

Answer

To solve the expression \( 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) \), we will use the properties of inverse trigonometric functions and some algebraic manipulations.

### Step-by-step Solution:

1. **Rewrite the Expression**:
\[ 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) = 2 \left( 2 \tan^{-1}\left(\frac{1}{5}\right) \right) - \tan^{-1}\left(\frac{1}{239}\right) \]

2. **Use the Double Angle Formula**:
The double angle formula for tangent states:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Let \( \theta = \tan^{-1}\left(\frac{1}{5}\right) \), then:
\[ \tan(2\theta) = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} = \frac{\frac{2}{5}}{1 - \frac{1}{25}} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2 \cdot 25}{5 \cdot 24} = \frac{10}{24} = \frac{5}{12} \]
Thus, we have:
\[ 2 \tan^{-1}\left(\frac{1}{5}\right) = \tan^{-1}\left(\frac{5}{12}\right) \]

3. **Substitute Back**:
Now substitute back into the expression:
\[ 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) = 2 \tan^{-1}\left(\frac{5}{12}\right) - \tan^{-1}\left(\frac{1}{239}\right) \]

4. **Use the Formula for the Difference of Arctangents**:
The formula for the difference of two arctangents is:
\[ \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left(\frac{x - y}{1 + xy}\right) \]
Here, let \( x = \frac{5}{12} \) and \( y = \frac{1}{239} \):
\[ \tan^{-1}\left(\frac{\frac{5}{12} - \frac{1}{239}}{1 + \frac{5}{12} \cdot \frac{1}{239}}\right) \]

5. **Calculate the Numerator**:
Find a common denominator for the numerator:
\[ \frac{5}{12} - \frac{1}{239} = \frac{5 \cdot 239 - 12 \cdot 1}{12 \cdot 239} = \frac{1195 - 12}{12 \cdot 239} = \frac{1183}{12 \cdot 239} \]

6. **Calculate the Denominator**:
Calculate the denominator:
\[ 1 + \frac{5}{12} \cdot \frac{1}{239} = 1 + \frac{5}{12 \cdot 239} = \frac{12 \cdot 239 + 5}{12 \cdot 239} = \frac{2868 + 5}{12 \cdot 239} = \frac{2873}{12 \cdot 239} \]

7. **Combine the Results**:
Now combine the results:
\[ \tan^{-1}\left(\frac{\frac{1183}{12 \cdot 239}}{\frac{2873}{12 \cdot 239}}\right) = \tan^{-1}\left(\frac{1183}{2873}\right) \]

8. **Final Result**:
Since \( \tan^{-1}(1) = \frac{\pi}{4} \), we find that:
\[ 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]

### Conclusion:
Thus, the final answer is:
\[ \frac{\pi}{4} \]