$\cot \left(\cos e{c}^{-1}\frac{5}{3}+{\tan }^{-1}\frac{2}{3}\right)$is

$$\begin{gathered} \cot \left(\cos e{c}^{-1}\frac{5}{3}+{\tan }^{-1}\frac{2}{3}\right)=cot\left({\tan }^{-1}\frac{3}{4}+{\tan }^{-1}\frac{2}{3}\right) \\ =cot\left[{\tan }^{-1}\left(\frac{{\displaystyle \frac{3}{4}}+{\displaystyle \frac{2}{3}}}{1-\frac{3}{4}\frac{2}{3}}\right)\right] \\ =cot\left({\tan }^{-1}\frac{17}{6}\right) \\ =cot\left(co{t}^{-1}\frac{6}{17}\right) \\ =\frac{6}{17}\cong 0.35 \end{gathered}$$