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

Options

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

Answer

To solve the expression \( \tan\left(2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) \), we will 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: Substitute into the double angle formula
Now substituting \( \tan(\theta) \) into the formula:
\[ \tan(2\tan^{-1}\left(\frac{1}{5}\right)) = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} \]

### Step 3: Simplify the expression
Calculating the denominator:
\[ 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25} = \frac{25 - 1}{25} = \frac{24}{25} \]
Now substituting back:
\[ \tan(2\tan^{-1}\left(\frac{1}{5}\right)) = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2}{5} \cdot \frac{25}{24} = \frac{50}{120} = \frac{5}{12} \]

### Step 4: Now we need to compute \( \tan\left(\frac{5}{12} - \frac{\pi}{4}\right) \)
Using the formula for \( \tan(a - b) \):
\[ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \]
Here, \( a = \tan^{-1}\left(\frac{5}{12}\right) \) and \( b = \tan^{-1}(1) \) (since \( \tan\left(\frac{\pi}{4}\right) = 1 \)).

### Step 5: Substitute into the tangent subtraction formula
We have:
\[ \tan\left(2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) = \frac{\frac{5}{12} - 1}{1 + \frac{5}{12} \cdot 1} \]

### Step 6: Simplify the numerator and denominator
Calculating the numerator:
\[ \frac{5}{12} - 1 = \frac{5}{12} - \frac{12}{12} = \frac{5 - 12}{12} = \frac{-7}{12} \]
Calculating the denominator:
\[ 1 + \frac{5}{12} = \frac{12}{12} + \frac{5}{12} = \frac{17}{12} \]

### Step 7: Final calculation
Now substituting back:
\[ \tan\left(2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) = \frac{\frac{-7}{12}}{\frac{17}{12}} = \frac{-7}{17} \]

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