# How do you evaluate cos(Arctan 2)?

Set $x = \arctan 2$ hence $\tan x = 2$ square both sides to get

${\tan}^{2} x = 4 \implies {\tan}^{2} x + 1 = 5 \implies \frac{1}{\cos} ^ 2 x = 5$

Hence $\cos x = \frac{1}{\sqrt{5}}$ or $x = \arccos \left(\frac{1}{\sqrt{5}}\right)$

But $x = \arctan 2$ hence

$\arctan 2 = \arccos \left(\frac{1}{\sqrt{5}}\right) \implies \cos \left(\arctan 2\right) = \cos \left(\arccos \left(\frac{1}{\sqrt{5}}\right)\right) = \frac{1}{\sqrt{5}}$

Finally $\cos \left(\arctan 2\right) = \frac{1}{\sqrt{5}}$