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

Options

A. `5/12`
B. `-5/12`
C. `7/17`
D. `-7/17`

Answer

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

### Step 1: Let \( x = 2\tan^{-1}\left(\frac{1}{5}\right) \)
This means that \( \tan^{-1}\left(\frac{1}{5}\right) = \frac{x}{2} \).

### Step 2: Find \( \tan(x) \)
Using the double angle formula for tangent, we have:
\[ \tan(x) = \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
where \( \theta = \tan^{-1}\left(\frac{1}{5}\right) \). Thus, \( \tan(\theta) = \frac{1}{5} \).

### Step 3: Substitute \( \tan(\theta) \) into the formula
Substituting \( \tan(\theta) \):
\[ \tan(x) = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} \]
Calculating \( \left(\frac{1}{5}\right)^2 = \frac{1}{25} \):
\[ \tan(x) = \frac{\frac{2}{5}}{1 - \frac{1}{25}} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2}{5} \cdot \frac{25}{24} = \frac{10}{24} = \frac{5}{12} \]

### Step 4: Now find \( \tan\left(x - \frac{\pi}{4}\right) \)
Using the tangent subtraction formula:
\[ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \]
where \( a = x \) and \( b = \frac{\pi}{4} \). We know \( \tan\left(\frac{\pi}{4}\right) = 1 \):
\[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan(x) - 1}{1 + \tan(x) \cdot 1} \]
Substituting \( \tan(x) = \frac{5}{12} \):
\[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\frac{5}{12} - 1}{1 + \frac{5}{12}} = \frac{\frac{5}{12} - \frac{12}{12}}{1 + \frac{5}{12}} = \frac{\frac{-7}{12}}{\frac{17}{12}} = \frac{-7}{17} \]

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