# How do you simplify the expression cos(arctan(x/5)) ?

Nov 10, 2016

cos(arctan(x/5))=+-root2(1/(1+(x/5)^2)

#### Explanation:

From the fundamental identity of trigonometry ${\cos}^{2} \theta + {\sin}^{2} \theta = 1$
we can deduce by dividing both sides for ${\cos}^{\theta}$ and imposing the existence condition $\theta \ne \frac{\pi}{2} + k \pi$

$1 + {\tan}^{2} \theta = \frac{1}{\cos} ^ 2 \theta$ that can be rewritten as

${\cos}^{2} \theta = \frac{1}{1 + {\tan}^{2} \theta}$ or

costheta=+-root2(1/(1+tan^2theta)
if $\theta$ is $\arctan \left(\frac{x}{5}\right)$ then
cos(arctan(x/5))=+-root2(1/(1+tan^2(arctan(x/5))
but $\tan \left(\arctan \left(\frac{x}{5}\right)\right) = \frac{x}{5}$
in the end we can finally write
cos(arctan(x/5))=+-root2(1/(1+(x/5)^2)