Iterative Solution Using Casio Calculator

OOOoooooommmmm....

https://www.efunda.com/formulae/solid_mechanics/columns/eccentric.cfm
http://www.mitcalc.com/doc/buckling/help/en/buckling.htm

Many Casio calculators do have trigonometric functions. You could do a Taylor series of the secant function to calculate the secant. There's also the obvious fact that you have internet access. There are many trigonometric calculating web pages.

There are a million (transcendental exaggeration) Casio calculators.

A little specificity will help.

Solving equations with the CASIO FX-991ES calculator - The Student

Can't you do it on XL? You need to go XL options/formulas and check the Allow iterative calculation box (otherwise you'll get a circular reference message).

Why are you going through all that trouble? Surely you know that the secant is simply the reciprocal of the cosine, so use the cosine function on your factors and then use the reciprocal key on the result for your answer.

It is good to know about the Taylor series. It is the iterative calculation of the Taylor series that most trigonometric calculators do in the background when one presses those trig keys.

What equation are trying to solve? From a quick look on the web the equation I found can be solved directly, as per RAMconsult in #5.

You only need iteration if you have θ and sec θ in the same equation (and you're trying to solve for θ)