Answer
$\frac{11\sqrt 5}{25}$
Work Step by Step
Recall that $\cos(A-B)=\cos A\cos B+\sin A\sin B$
$\implies \cos(\arcsin\frac{4}{5}-\arctan 2)=\cos (\arcsin \frac{4}{5})\cos(\arctan 2)+\sin(\arcsin \frac{4}{5})\sin(\arctan 2)$
$=\sqrt {1-(\frac{4}{5})^{2}}\cdot \frac{1}{\sqrt {1+2^{2}}}+\frac{4}{5}\cdot\frac{2}{\sqrt {1+2^{2}}}$
$=\frac{3}{5}\cdot\frac{\sqrt 5}{5}+\frac{8\sqrt 5}{25}$
$=\frac{11\sqrt 5}{25}$
