Let \(x = \tan^{-1}\frac{1}{2}\). Then, \(\tan x = \frac{1}{2}\).

We need to find \(\sec^2 x\). We know that \(\sec^2 x = 1 + \tan^2 x\).

So, \(\sec^2(\tan^{-1}\frac{1}{2}) = 1 + (\frac{1}{2})^2 = 1 + \frac{1}{4} = \frac{5}{4}\).

Let \(y = \cot^{-1}\frac{1}{3}\). Then, \(\cot y = \frac{1}{3}\).

We need to find \(\csc^2 y\). We know that \(\csc^2 y = 1 + \cot^2 y\).

So, \(\csc^2(\cot^{-1}\frac{1}{3}) = 1 + (\frac{1}{3})^2 = 1 + \frac{1}{9} = \frac{10}{9}\).

Therefore, \(\sec^{2}(\tan^{-1}\frac{1}{2})+cosec^{2}(\cot^{-1}\frac{1}{3}) = \frac{5}{4} + \frac{10}{9}\).

To add these fractions, we find a common denominator, which is 36.

\(\frac{5}{4} + \frac{10}{9} = \frac{5 \times 9}{4 \times 9} + \frac{10 \times 4}{9 \times 4} = \frac{45}{36} + \frac{40}{36} = \frac{85}{36}\).