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 $\cos(\text{atan}(2))$
 A: You can think of $\arctan(2)$ as an angle whose tangent is $2$.  
To visualize such an angle, draw a right triangle and mark an acute angle.  Then opposite the marked angle have a side (leg) of length $2$, and adjacent to the marked angle have a side (leg) of length $1$.  
The marked angle now represents $\arctan(2)$.  
You can then fill in the hypotenuse of the right triangle, and use your diagram to find the necessary values for your problem.
A: Hint: $\sin(\arctan(2))=\frac{\tan(\arctan(2))}{\sec(\arctan(2))}=\frac2{\sqrt5}$ and $\cos(\arctan(2))=\frac1{\sec(\arctan(2))}=\frac1{\sqrt5}$
Recall that $\sec^2(\theta)=\tan^2(\theta)+1$
A: Graphically, we may say $\theta$ is an angle in a right-angled triangle where the opposite side is 2 and the adjacent side is 1, so that $\arctan{\theta}=2$
As the Hypotenuse side=$\sqrt5$, we have $\cos{(\arctan2)}=\frac{1}{\sqrt5}$

A: 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
$\cos\theta=\cos(\arctan2)=1/\sqrt5$.
