Answer
$\dfrac{2\sqrt{5}}{5}$
Work Step by Step
Let $\alpha = arc\sin{\dfrac{3}{5}} \hspace{30pt} \beta = arc\tan{2}$
$\cos{(\alpha- \beta)} = \cos{\alpha} \cos{\beta} + \sin{\alpha} \sin{\beta} $
$\sin{\alpha} = \dfrac{3}{5}$
$\cos{\alpha} = \sqrt{1-\sin^2{\alpha}} = \dfrac{4}{5}$
$\tan{\beta} = 2 $
$\sec{\beta} = \sqrt{1+\tan^2{\beta}} = \sqrt{5}$
$\cos{\beta} = \dfrac{1}{\sec{\beta}} = \dfrac{\sqrt{5}}{5}$
$\sin{\beta} = \sqrt{1-\cos^2{\alpha}} = \dfrac{2\sqrt{5}}{5}$
$\cos{(\alpha- \beta)} = (\dfrac{4}{5})(\dfrac{\sqrt{5}}{5})+(\dfrac{3}{5})(\dfrac{2\sqrt{5}}{5})$
$\cos{(\alpha- \beta)} = \dfrac{2\sqrt{5}}{5}$
