Hello, overduex!
Find the exact value of: .$\displaystyle \sec\left[\arctan\left(\frac{3}{5}\right)\right]$
You're thinking is correct . . . up to a point.
Let $\displaystyle \theta \,=\,\arctan\left(\frac{3}{5}\right)\quad\Rightarrow\quad\tan\theta \:=\:\frac{3}{5}$
Tangent is negative in Quadrants 2 and 4.
There are two possible positions for angle $\displaystyle \theta.$
Code:

* 
: * √34 
3: * 
: *  5
  +      +      +  
5  * :
 * :3
 √34 * :
 *

Hence, there are two possible values for the secant.
. . $\displaystyle \sec\theta \;=\;\pm\frac{\sqrt{34}}{5}$