The value of $\tan^2(\sec^{-1} 2) + \cot^2(\cosec^{-1} 3)$ is equal to

Let $\theta = \sec^{-1} 2$

This means $\sec \theta = 2$

Since $\sec \theta = \frac{1}{\cos \theta}$:

$\cos \theta = \frac{1}{2}$

Using the identity $\sec^2 \theta - \tan^2 \theta = 1$:

$\tan^2 \theta = \sec^2 \theta - 1$

$\tan^2 \theta = (2)^2 - 1$

$\tan^2 \theta = 4 - 1$

$\tan^2 \theta = 3$

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Let $\phi = \cosec^{-1} 3$

This means $\cosec \phi = 3$

Since $\cosec \phi = \frac{1}{\sin \phi}$:

$\sin \phi = \frac{1}{3}$

Using the identity $\cosec^2 \phi - \cot^2 \phi = 1$:

$\cot^2 \phi = \cosec^2 \phi - 1$

$\cot^2 \phi = (3)^2 - 1$

$\cot^2 \phi = 9 - 1$

$\cot^2 \phi = 8$

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$\tan^2(\sec^{-1} 2) + \cot^2(\cosec^{-1} 3) = 3 + 8$

$= 11$

Therefore, the value is $11$.