[Pre-cal] How do you convert a hyperbolic equation into a parametric form?

Textbook asks us to graph the hyperbola using a parametric grapher, however, they don't explain how to convert.

I know to convert something like an ellipses or circle, you use:

x = acos(t)
y = bsin(t)

I know for this case you would set sin^2(t) = x^2 / a^2 to convert, and that makes sense, but I'm not sure why you would use:

x = asec(t)
y = btan(t)

Is there an intuitive way to understand why we're using sec and tan?

EDIT
Here's the equation to graph and convert:

(y^2 /25) - (x^4 / 4) = 1