Evaluate : \[\sec ^{2}\left(\tan ^{-1} \frac{1}{2}\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} \frac{1}{3}\right) \]

$$\begin{aligned} & \sec ^{2}\left(\tan ^{-1} \frac{1}{2}\right)+\operatorname{cosec}\left(\cot ^{-1} \frac{1}{3}\right) \\ & =\left[1+\tan ^{2}\left(\tan ^{-1} \frac{1}{2}\right)\right]+\left[1+\cot ^{2}\left(\cot ^{-1} \frac{1}{3}\right)\right] \\ & =\left[1+\left(\frac{1}{2}\right)^{2}\right]+\left[1+\left(\frac{1}{3}\right)^{2}\right] \\ & =\frac{85}{36} \end{aligned} $$