Question

The value of (\tan [2 \tan^{-1} \frac{1}{5} - \frac{\pi}{4}]) is

• A. (\frac{5}{4})
• B. (\frac{5}{16})
• C. (\frac{-7}{17})
• D. [suspicious link removed]

Hint:

$2 \tan^{-1} x = \tan^{-1}\left(\frac{2x}{1-x^2}\right)$ for $|x| < 1$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to evaluate a composition of trigonometric and inverse trigonometric functions.
Step 2: Key Formula or Approach:
1) \(2 \tan^{-1} x = \tan^{-1} (\frac{2x}{1-x^2})\) for \(|x|<1\).
2) \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).
Step 3: Detailed Explanation:
Let \(A = 2 \tan^{-1} \frac{1}{5}\).
Using the formula:
\[ \tan A = \tan(2 \tan^{-1} \frac{1}{5}) = \frac{2(1/5)}{1 - (1/5)^2} = \frac{2/5}{1 - 1/25} = \frac{2/5}{24/25} = \frac{2}{5} \times \frac{25}{24} = \frac{5}{12} \]
Now we need to find \(\tan(A - \frac{\pi}{4})\).
Using the subtraction formula for tangent:
\[ \tan(A - \frac{\pi}{4}) = \frac{\tan A - \tan(\pi/4)}{1 + \tan A \tan(\pi/4)} \]
Substitute \(\tan A = \frac{5}{12}\) and \(\tan(\pi/4) = 1\):
\[ = \frac{\frac{5}{12} - 1}{1 + \frac{5}{12} \times 1} = \frac{\frac{5-12}{12}}{\frac{12+5}{12}} = \frac{-7/12}{17/12} = -\frac{7}{17} \]
Step 4: Final Answer:
The value is \(-\frac{7}{17}\).