Question

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

Answer 1

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

Explanation:

From the fundamental identity of trigonometry `cos^2theta+sin^2theta=1`
we can deduce by dividing both sides for `cos^theta` and imposing the existence condition `theta ne pi/2+kpi`

`1+tan^2theta=1/cos^2theta` that can be rewritten as

`cos^2theta=1/(1+tan^2theta)` or

`costheta=+-root2(1/(1+tan^2theta)`
if `theta` is `arctan (x/5)` then
`cos(arctan(x/5))=+-root2(1/(1+tan^2(arctan(x/5))`
but `tan(arctan(x/5))=x/5`
in the end we can finally write
`cos(arctan(x/5))=+-root2(1/(1+(x/5)^2)`