$\tan^2(\sec^{-1} 2) + \cot^2(\csc^{-1} 3).$

This problem asks you to evaluate expressions involving inverse trig functions—the key is to construct a right triangle for each inverse function, then read off the sides you need. For $\sec^{-1} 2$, ask "what angle has secant equal to 2?" and sketch a triangle where the hypotenuse is 2 and the adjacent side is 1. Similarly, for $\csc^{-1} 3$, build a triangle with hypotenuse 3 and opposite side 1. Once you have the triangles, use the Pythagorean theorem to find the missing sides, then compute $\tan^2$ and $\cot^2$ directly from those side lengths.

This problem asks you to evaluate expressions involving inverse trig functions—the key is to construct a right triangle for each inverse function, then read off the sides you need. For $\sec^{-1} 2$, ask "what angle has secant equal to 2?" and sketch a triangle where the hypotenuse is 2 and the adjacent side is 1. Similarly, for $\csc^{-1} 3$, build a triangle with hypotenuse 3 and opposite side 1. Once you have the triangles, use the Pythagorean theorem to find the missing sides, then compute $\tan^2$ and $\cot^2$ directly from those side lengths.