Question

How do you find the amplitude and period of `Arc Sec`?

Answer 1

arcsec does not have a (meaningful) amplitude or period.
You seem to have confused arcsec with the direct trigonometric functions

Explanation:

arcsec(x) is an angle; typically its value is restricted to angles in the range `[0,2pi)`

In terms of a unit circle arcsec(x) is the angle `theta` such that the hypotenuse divided by the adjacent side of a unit circle triangle based at the origin has the ratio `x`.
Since the hypotenuse must be at least as long as any other side `x` is undefined for any value in the range `(-1,+1)`

The graph below demonstrates the arcsin relationship.
The vertical axis is the angle (in radians).
The horizontal axis is the arcsin argument value.

graph{x=sec(y) [-14.24, 14.23, -7.11, 7.13]}