The value of $ sin\left(2 tan^{-1}\frac{1}{3}\right) + cos \left(tan^{-1}2\sqrt{2}\right)$, is

$\frac{14}{15}$

$ sin\left(2 tan^{-1}\frac{1}{3}\right) + cos \left(tan^{-1}2\sqrt{2}\right)$

$=sin \left(sin^{-1}\frac{3}{5}\right) + cos \left(cos^{-1}\frac{1}{3}\right) $     $\left[∵ 2 tan^{-1}x = sin^{-1}\frac{2x}{1+x^2}\right]$

$=\frac{3}{5}+\frac{1}{3}=\frac{14}{15}$