The value of (cosec^ -1 (5)/(3)+ ^ -1 (2)/(3)) is

\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}