The solution set of inequality  $({\cot }^{-1}\mathrm{x})({\tan }^{-1}\mathrm{x})+(2-\frac{\mathrm{\pi }}{2})\times {\cot }^{-1}\mathrm{x}-3{\tan }^{-1}\mathrm{x}-3(2-\frac{\mathrm{\pi }}{2})>0$ is (a, b), then the value  of ${\cot }^{-1}\mathrm{a}+{\cot }^{-1}\mathrm{b}$ is

$\begin{array}{rl} & ({\cot }^{-1}\mathrm{x})({\tan }^{-1}\mathrm{x})+(2-\frac{\mathrm{\pi }}{2}){\cot }^{-1}\mathrm{x}-3{\tan }^{-1}\mathrm{x}-3(2-\frac{\mathrm{\pi }}{2})>0\\ & \Rightarrow {\cot }^{-1}\mathrm{x}({\tan }^{-1}\mathrm{x}-\frac{\mathrm{\pi }}{2})+2{\cot }^{-1}\mathrm{x}-6-3({\tan }^{-1}\mathrm{x}-\frac{\mathrm{\pi }}{2})>0\\ & \Rightarrow -{({\cot }^{-1}\mathrm{x})}^2+5{\cot }^{-1}\mathrm{x}-6>0\\ & \Rightarrow ({\cot }^{-1}\mathrm{x}-3)(2-{\cot }^{-1}\mathrm{x})>0\\ & \Rightarrow ({\cot }^{-1}\mathrm{x}-3)({\cot }^{-1}\mathrm{x}-2)<0\\ & \Rightarrow 2<{\cot }^{-1}\mathrm{x}<3\\ & \Rightarrow \cot 3<\mathrm{x}<\cot 2[{\mathrm{ascot}}^{-1}\mathrm{x}\text{is a decreasing function}]\\ & \Rightarrow \text{Hence,}\mathrm{x}\in (\cot 3,\cot 2)\\ & \Rightarrow {\cot }^{-1}\mathrm{a}+{\cot }^{-1}\mathrm{b}={\cot }^{-1}(\cot 3)+{\cot }^{-1}(\cot 2)=5\end{array}$