MHT CET · Maths · Inverse Trigonometric Functions
\(\tan \left(\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right)=\)
- A \(\frac{17}{6}\)
- B \(\frac{17}{3}\)
- C \(\frac{18}{5}\)
- D \(\frac{7}{15}\)
Answer & Solution
Correct Answer
(A) \(\frac{17}{6}\)
Step-by-step Solution
Detailed explanation
\(\tan \left[\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right] \)
\( =\tan \left[\tan ^{-1}\left(\frac{3}{4}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]=\) \(\tan \left[\tan ^{-1}\left(\frac{\left(\frac{3}{4}\right)+\left(\frac{2}{3}\right)}{1-\left(\frac{3}{4}\right)+\left(\frac{2}{3}\right)}\right)\right] \)
\( =\tan \left[\tan ^{-1}\left(\frac{17}{6}\right)\right]=\frac{17}{6}\)
\( =\tan \left[\tan ^{-1}\left(\frac{3}{4}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]=\) \(\tan \left[\tan ^{-1}\left(\frac{\left(\frac{3}{4}\right)+\left(\frac{2}{3}\right)}{1-\left(\frac{3}{4}\right)+\left(\frac{2}{3}\right)}\right)\right] \)
\( =\tan \left[\tan ^{-1}\left(\frac{17}{6}\right)\right]=\frac{17}{6}\)
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