# How do you simplify sec(arctan(4x))?

Refer to explanation

#### Explanation:

Set $t = \arctan 4 x \implies \tan t = 4 x = \frac{4 x}{1}$

Now imagine a right triangle with perpendicular sides $1 , 4 x$
hence from Pythagorean theorem the hypotenuse is

$c = \sqrt{{1}^{2} + {\left(4 x\right)}^{2}} \implies c = \sqrt{1 + 16 {x}^{2}}$

Finally we have that

$\sec \left(\arctan 4 x\right) = \sec \left(t\right) = \frac{\sqrt{1 + 16 {x}^{2}}}{1} = \sqrt{1 + 16 {x}^{2}}$