(C) Let $\theta = \tan ^{-1} 2$,then $\tan \theta = 2$. We know that $\sec ^2 \theta = 1 + \tan ^2 \theta$. Thus,$\sec ^2(\tan ^{-1} 2) = 1 + (2)^2 =

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(C) Let $\theta = \tan ^{-1} 2$,then $\tan \theta = 2$. We know that $\sec ^2 \theta = 1 + \tan ^2 \theta$. Thus,$\sec ^2(\tan ^{-1} 2) = 1 + (2)^2 = 1 + 4 = 5$. Let $\phi = \cot ^{-1} 3$,then $\cot \phi = 3$. We know that $\operatorname{cosec}^2 \phi = 1 + \cot ^2 \phi$. Thus,$\operatorname{cosec}^2(\cot ^{-1} 3) = 1 + (3)^2 = 1 + 9 = 10$. Adding these values,we get $5 + 10 = 15$.