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

Let ${\tan }^{-1} 1/3=\alpha$ and ${\tan }^{-1} 2\sqrt{2}=\beta$. Then

$\tan \alpha =1/3$ and $\tan \beta =2\sqrt{2}$, so that

$\begin{array}{c}\sin \left(2{\tan }^{-1} (1/3)\right)+\cos \left({\tan }^{-1} 2\sqrt{2}\right)-\sin 2\alpha +\cos \beta \\ =\frac{2\tan \alpha }{1+{\tan }^2 \alpha }+\frac{1}{\sqrt{1+{\tan }^2 \beta }}[\because -\pi /2<\beta <\pi /2]\\ =\frac{2\cdot (1/3)}{1+(1/9)}+\frac{1}{\sqrt{1+8}}=\frac{2}{3}\cdot \frac{9}{10}+\frac{1}{3}=\frac{3}{5}+\frac{1}{3}=\frac{14}{15}\end{array}$