Answer
$\dfrac{2 \sqrt{15} -\sqrt{5}}{10}$
Work Step by Step
$\alpha = \tan^{-1} \dfrac{1}{2} \hspace{30pt} \beta = \sin^{-1} \dfrac{1}{2}$
$\tan{\alpha} = \dfrac{1}{2}$
$\sec{\alpha} = \sqrt{1+\tan^2{\alpha}} = \dfrac{\sqrt{5}}{2}$
$\cos{\alpha} = \dfrac{1}{\sec{\alpha}} = \dfrac{2\sqrt{5}}{5}$
$\sin{\alpha} = \sqrt{1-\cos^2{\alpha}} = \dfrac{\sqrt{5}}{5}$
$\sin{\beta} = \dfrac{1}{2}$
$\cos{\beta} =\sqrt{1-\sin^2{\beta}} = \dfrac{\sqrt{3}}{2}$
$\cos{(\alpha+\beta)} = \cos{\alpha} \cos{\beta} - \sin{\alpha} \sin{\beta} $
$\cos{(\alpha+\beta)} = (\dfrac{2\sqrt{5}}{5})(\dfrac{\sqrt{3}}{2})-(\dfrac{1}{2})(\dfrac{\sqrt{5}}{5})$
$\cos{(\alpha+\beta)} = \dfrac{2 \sqrt{15} -\sqrt{5}}{10}$
