Q62 / 105 · JEE Main 2008 · Mathematics
The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)$ is
- Correct answerA$\frac{6}{17}$
- B$\frac{3}{17}$
- C$\frac{4}{17}$
- D$\frac{5}{17}$
Solution
$$
\begin{aligned}
& \text { Let } E=\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right) \\
& \Rightarrow E=\cot \left(\tan ^{-1}\left(\frac{3}{4}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right) \\
& \Rightarrow E=\cot \left(\tan ^{-1}\left(\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4} \cdot \frac{2}{3}}\right)\right) \\
& \Rightarrow E=\cot \left(\tan ^{-1} \frac{17}{6}\right)=\frac{6}{17} .
\end{aligned}
$$
Hence, the correct answer is option: 'A'.