The value of $\sec ^{2}\left(\tan ^{-1} 2\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} 3\right)$ is ____________________.
We know
$\tan ^{-1} x=\sec ^{-1} \sqrt{1+x^{2}}$ and $\cot ^{-1} x=\operatorname{cosec}^{-1} \sqrt{1+x^{2}}$
So,
$\tan ^{-1} 2=\sec ^{-1} \sqrt{1+2^{2}}=\sec ^{-1} \sqrt{5}$
$\cot ^{-1} 3=\operatorname{cosec}^{-1} \sqrt{1+3^{2}}=\operatorname{cosec}^{-1} \sqrt{10}$
$\therefore \sec ^{2}\left(\tan ^{-1} 2\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} 3\right)$
$=\sec ^{2}\left(\sec ^{-1} \sqrt{5}\right)+\operatorname{cosec}^{2}\left(\operatorname{cosec}^{-1} \sqrt{10}\right)$
$=\left[\sec \left(\sec ^{-1} \sqrt{5}\right)\right]^{2}+\left[\operatorname{cosec}\left(\operatorname{cosec}^{-1} \sqrt{10}\right)\right]^{2}$
$=(\sqrt{5})^{2}+(\sqrt{10})^{2}$
$=15$\
Thus, the value of $\sec ^{2}\left(\tan ^{-1} 2\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} 3\right)$ is 15 .
The value of $\sec ^{2}\left(\tan ^{-1} 2\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} 3\right)$ is ____15____.
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