The values of $sec^2(tan^{-1}2)+cosec^2(cot^{-1}3)= 15 $, is

15

We have, $sec^2(tan^{-1}2)+cosec^2(cot^{-1}3)$

$= \begin{Bmatrix}sec (tan^{-1}2)\end{Bmatrix}^2+\begin{Bmatrix}cosec (cot^{-1}3)\end{Bmatrix}^2$

$= \begin{Bmatrix}sec \left(tan^{-1}\frac{2}{1}\right)\end{Bmatrix}^2+\begin{Bmatrix}cosec \left(cot^{-1}\frac{3}{1}\right)\end{Bmatrix}^2$

$= \begin{Bmatrix} sec(sec^{-1}\sqrt{5})\end{Bmatrix}^2+ \begin{Bmatrix}cosec(cosec^{-1}\sqrt{10})\end{Bmatrix}^2$

$= (\sqrt{5})^2 + (\sqrt{10})^2 = 15 $