Notice we have:

$\displaystyle \alpha+\beta=\frac{\pi}{2}$

$\displaystyle \alpha=\tan^{-1}(2)$

$\displaystyle \beta=\sin^{-1}\left(\frac{1}{\sqrt{5}}\right)$

thus:

$\displaystyle \sin^{-1}\left(\frac{1}{\sqrt{5}}\right)+\tan^{-1}(2)=\frac{\pi}{2}$

Notice that $\displaystyle \gamma=\alpha-\beta$

$\displaystyle \gamma=\tan^{-1}(2)-\sin^{-1}\left(\frac{1}{\sqrt{5}}\right)$

Using the law of cosines allows you to find $\displaystyle \gamma$.

$\displaystyle \gamma=\cos^{-1}\left(\frac{4}{5}\right)=\sin^{-1}\left(\frac{3}{5}\right)=\tan^{-1}\left(\frac{3}{4}\right)$