How to analytically evaluate $\cos(\pi/4+\text{atan}(2))$ This equals $\cos(\pi/4)\cos(\text{atan}(2))-\sin(\pi/4)\sin(\text{atan}(2))$.
I'm just not sure how to evaluate $\cos(\text{atan}(2))$

Here's another explanation.  Let $\theta=\arctan 2$ so that $\tan\theta=2$.  Since, $\sec\theta=\sqrt{1+\tan^2}\theta$, we get that $\sec\theta=\sqrt5$; hence
$\cos\theta=\cos(\arctan2)=1/\sqrt5$.
