# How do you find the exact value of cos[arctan(-5/12)]?

Notice that 5 and 12 are two sides of a right angled triangle whose hypotenuse is 13, since ${5}^{2} + {12}^{2} = 25 + 144 = 169 = {13}^{2}$
So if $\theta$ is the smallest angle in the $5$,$12$,$13$ triangle then
$\sin \theta = \frac{5}{13}$, $\cos \theta = \frac{12}{13}$ and $\tan \theta = \frac{5}{12}$.
Then $\tan \left(- \theta\right) = - \tan \left(\theta\right) = - \frac{5}{12}$
So we are looking for $\cos \left(- \theta\right) = \cos \left(\theta\right) = \frac{12}{13}$