The values of ${\sec }^2\left({\tan }^{-1}2\right)+{\mathrm{cosec}}^2\left({\cot }^{-1}3\right)=$

Solution:

We have,

$\begin{array}{l}{\sec }^2\left({\tan }^{-1}2\right)+{\mathrm{cosec}}^2\left({\cot }^{-1}3\right)\\ ={\left\{\sec \left({\tan }^{-1}2\right)\right\}}^2+{\left\{\mathrm{cosec}\left({\cot }^{-1}3\right)\right\}}^2\\ ={\left\{\sec \left({\tan }^{-1}\frac{2}{1}\right)\right\}}^2+{\left\{\mathrm{cosec}\left({\cot }^{-1}\frac{3}{1}\right)\right\}}^2\\ ={\left\{\sec \left({\sec }^{-1}\sqrt{5}\right)\right\}}^2+{\left\{\mathrm{cosec}\left({\mathrm{cosec}}^{-1}\sqrt{10}\right)\right\}}^2\\ =(\sqrt{5}{)}^2+(\sqrt{10}{)}^2=15\end{array}$