question_answer

Find the value of \[{{\cos }^{-1}}\left( \frac{4}{5} \right)+{{\tan }^{-1}}\left( \frac{3}{5} \right)\].

A)  \[{{\tan }^{-1}}\left( \frac{7}{10} \right)\]

B)  \[{{\tan }^{-1}}\left( \frac{27}{11} \right)\]

C)  \[{{\sin }^{-1}}\left( \frac{7}{5} \right)\]

D)  \[{{\tan }^{-1}}\left( \frac{1}{5} \right)\]

Correct Answer: B

Solution :

\[{{\cos }^{-1}}\left( \frac{4}{5} \right)+{{\tan }^{-1}}\left( \frac{3}{5} \right)\] \[\Rightarrow \] \[{{\tan }^{-1}}\left( \frac{3}{4} \right)+{{\tan }^{-1}}\left( \frac{3}{5} \right)\] \[={{\tan }^{-1}}\left\{ \frac{\frac{3}{4}+\frac{3}{5}}{2-\frac{3}{4}.\frac{3}{5}} \right\}={{\tan }^{-1}}\left\{ \frac{\frac{15\times 12}{20}}{\frac{20-9}{20}} \right\}\] \[={{\tan }^{-1}}\left( \frac{27}{11} \right)\]