MHT CET · Maths · Inverse Trigonometric Functions
The value of \(\tan \left[2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right]\) is
- A \(\frac{5}{4}\)
- B \(\frac{5}{16}\)
- C \(-\frac{7}{17}\)
- D \(\frac{7}{17}\)
Answer & Solution
Correct Answer
(C) \(-\frac{7}{17}\)
Step-by-step Solution
Detailed explanation
\(2 \tan^{-1} \frac{1}{5} = \tan^{-1} \left(\frac{2 \cdot \frac{1}{5}}{1-(\frac{1}{5})^2}\right) = \tan^{-1} \left(\frac{2/5}{24/25}\right) = \tan^{-1} \frac{5}{12}\) \(\tan \left[\tan^{-1} \frac{5}{12} - \frac{\pi}{4}\right] = \frac{\frac{5}{12} - \tan \frac{\pi}{4}}{1 + \frac{5}{12} \tan \frac{\pi}{4}} = \frac{\frac{5}{12} - 1}{1 + \frac{5}{12} \cdot 1} = \frac{-7/12}{17/12} = -\frac{7}{17}\)
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