The value of tan `(2 "tan"^(-1)(1)/(5)-(pi)/(4))` is

Options

A. `-(7)/(17)`
B. `(5)/(16)`
C. `(5)/(4)`
D. `(7)/(17)`

Explanation

To find the value of \( \tan\left(2 \tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) \), we can follow these steps:

### Step 1: Use the double angle formula for tangent
We know that:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Let \( \theta = \tan^{-1}\left(\frac{1}{5}\right) \). Therefore, \( \tan(\theta) = \frac{1}{5} \).

### Step 2: Calculate \( \tan(2\theta) \)
Using the double angle formula:
\[ \tan(2\theta) = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} \]
Calculating the denominator:
\[ 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25} = \frac{25 - 1}{25} = \frac{24}{25} \]
Now substituting back:
\[ \tan(2\theta) = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2}{5} \cdot \frac{25}{24} = \frac{50}{120} = \frac{5}{12} \]

### Step 3: Find \( \tan(2\theta - \frac{\pi}{4}) \)
Using the tangent subtraction formula:
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
Here, \( a = 2\theta \) and \( b = \frac{\pi}{4} \), where \( \tan\left(\frac{\pi}{4}\right) = 1 \):
\[ \tan(2\theta - \frac{\pi}{4}) = \frac{\tan(2\theta) - 1}{1 + \tan(2\theta) \cdot 1} \]
Substituting \( \tan(2\theta) = \frac{5}{12} \):
\[ \tan(2\theta - \frac{\pi}{4}) = \frac{\frac{5}{12} - 1}{1 + \frac{5}{12}} = \frac{\frac{5}{12} - \frac{12}{12}}{1 + \frac{5}{12}} = \frac{\frac{-7}{12}}{\frac{17}{12}} = \frac{-7}{17} \]

### Conclusion
Thus, the value of \( \tan\left(2 \tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) \) is:
\[ \boxed{-\frac{7}{17}} \]