Question

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

• A. \(\frac{17}{7}\)
• B. \(-\frac{17}{7}\)
• C. \(\frac{7}{17}\)
• D. \(-\frac{7}{17}\)

Hint:

First simplify \(2\tan^{-1}x\), then apply \(\tan(A-B)\).

Answer 1

The Correct Option is D

Concept: \[ \tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta} \quad,\quad \tan(A-B)=\frac{\tan A - \tan B}{1+\tan A \tan B} \]

Step 1: Let \(\theta=\tan^{-1}(1/5)\).
\[ \tan(2\theta) = \frac{2(1/5)}{1-(1/25)} = \frac{2/5}{24/25} = \frac{2}{5}\cdot\frac{25}{24} = \frac{5}{12} \]

Step 2: Use \(\tan(A-B)\).
\[ \tan\left(2\theta - \frac{\pi}{4}\right) = \frac{\frac{5}{12} - 1}{1 + \frac{5}{12}} \] \[ = \frac{-7/12}{17/12} = -\frac{7}{17} \] Conclusion: \[ {-\frac{7}{17}} \]