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: Calculate $2 \tan^{-1(1/5)$}
$\tan^{-1}\left(\frac{2/5}{1 - 1/25}\right) = \tan^{-1}\left(\frac{2/5}{24/25}\right) = \tan^{-1}\left(\frac{5}{12}\right)$.

Step 2: Use $\tan(A - B)$ formula
Let $A = \tan^{-1}(5/12)$ and $B = \pi/4$.
$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} = \frac{5/12 - 1}{1 + (5/12)(1)}$.

Step 3: Simplify
$\frac{-7/12}{17/12} = -\frac{7}{17}$.
Final Answer: (C)