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