L.H.S. $=\sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 3\right)$
$=\left[\sec \left(\tan ^{-1} 2\right)\right]^2+\left[\operatorname{cosec}\left(\cot ^{-1} 3\right)\right]^2$
$=\left[\sec \left(\sec ^{-1} \sqrt{5}\right)\right]^2+\left[\operatorname{cosec}\left(\operatorname{cosec}^{-1} \sqrt{10}\right)\right]^2$
$=(\sqrt{5})^2+(\sqrt{10})^2=5+10=15=$ R.H.S.
