$\begin{aligned} & \text { Let } \tan ^{-1} 2=\alpha \Rightarrow \tan \alpha=2 \\ & \quad \text { And } \cot ^{-1} 3=\beta \Rightarrow \cot \beta=3 \\ & \therefore \quad \sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 3\right) \\ & =\sec ^2 \alpha+\operatorname{cosec}^2 \beta \\ & =1+\tan ^2 \alpha+1+\cot ^2 \beta \\ & =2+(2)^2+(3)^2 \\ & =15\end{aligned}$