I have lately been working on trigonometric identities, which I am not very good at. I would like to create a program that can do these for me, or be able to tell me if I'm wrong. I don't want to include every theorem, axiom, identity, and everything else, because that would make this program exceedingly large and slow.

Any ideas on where I should start?

You could make it real easy with the basic menu function. In this link for example, it divides the identities into 10 parts. Since the menu has enough options for easy access, it shouldn't be hard to then display the formula, input values with variables, or have it stored to a string to use, etc.
Of course, if you want to take the long way and make a custom menu to select, same sort of organization probably, or you can make it fancy, your call.

Well, you can check your work for any identity already—just use some random angle that isn't likely to make things equal. For example, if you wanted to check the identity tan^2(x)+1=sec^2(x), you could plug into your calculator, tan^2(1)+1=sec^2(1).

How do I doe sin^2(X)? I don't have a sin^2( command on my calc, unless it's in some different format.

you do sin(x)^2
And I just use the nspire for trigonometric identities, it doesn't do all the work for me, but it definitely helps, it can simplify anything to sin/cos identities, which are obviously much easier to work with than csc/cot/sec... (well to me at least) but yeah I just play around with them till I get to the other side of the equation :') but this would be a great program to have on non-CAS calculators, definitely would be interested to see how this would be done, given the fact that there are like a zillion identities...

Ok... Time to get to work!