What is the value of $\sec^{2}\left(\tan^{-1} 2\right) + \operatorname{cosec}^{2}\left(\cot^{-1} 3\right)$?

Explanation

1. 01

   $\sec^{2} \theta = 1 + \tan^{2} \theta$, so $\sec^{2}\left(\tan^{-1} 2\right) = 1 + 4 = 5$.

2. 02

   $\operatorname{cosec}^{2} \phi = 1 + \cot^{2} \phi$, so $\operatorname{cosec}^{2}\left(\cot^{-1} 3\right) = 1 + 9 = 10$; the sum is 15.