$$\begin{gathered} {\text{If the solution set of inequality (cot}}^{-1}x) (ta{n}^{-1}x)+(2-\frac{\mathrm{\pi }}{2})co{t}^{-1}x-3ta{n}^{-1 }x-3(2-\frac{\mathrm{\pi }}{2})>0 is (a,b), \\ then the value of co{t}^{-1 }a+co{t}^{-1}b is \end{gathered}$$

$\begin{array}{rcl} ({\cot }^{-1}x) ({\tan }^{-1}x)+(2-\frac{\mathrm{\pi }}{2})co{t}^{-1}x-3{\tan }^{-1 }x-3(2-\frac{\mathrm{\pi }}{2})& >& 0\\ & \Rightarrow & {\cot }^{-1}x ({\tan }^{-1}x+2-\frac{\mathrm{\pi }}{2})-3({\tan }^{-1 }x+2-\frac{\mathrm{\pi }}{2})>0\\ & \Rightarrow & ({\cot }^{-1}x-3) ({\tan }^{-1}x+2-\frac{\mathrm{\pi }}{2})>0\\ S\mathrm{in}ce {\tan }^{-1}x-\frac{\mathrm{\pi }}{2} & =& -{\cot }^{-1}x, \\ & \Rightarrow & ({\cot }^{-1}x-3)(2-co{t}^{-1}x) >0\\ & \Rightarrow & ({\cot }^{-1}x-3)(co{t}^{-1}x-2) <0\\ & \Rightarrow & 2<{\cot }^{-1}x<3\\ & \Rightarrow & \cot 3<x<cot 2 \left(\sin ce it is decrea\sin g function\right)\\ Hence x& \in & (cot3,cot2) \\ co{t}^{-1 }a+co{t}^{-1}b& =& co{t}^{-1 }\left(cot3\right)+co{t}^{-1}\left(cot2\right)\\ & =& 3+2\\ & =& 5\end{array}$