There already is such a program for the ti 84, does exactly what you are saying. its somewhere on ticalc,  i just haven't looked at the 84 section in a while. anyway, i am in AP Calc and derivatives with trig really arent that hard.
d/dx(sin(x))=cos(x)
d/dx(cos(x))=-sin(x)
d/dx(tan(x))=sec^(x)
d/dx(cot(x))=-csc^2(x)
d/dx(sec(x)=sec(x)*tan(x)
d/dx(csc(x))=-csc(x)*cot(x)
I take it you have already programed in product, quotient and chain rules? if not, id start with those before worrying about trig. it would also be cool if this program could do integrals. as for combining like terms? i have no idea.

ok, here is that 84 app.
http://www.ticalc.org/archives/files/fileinfo/278/27805.html

I swapped my 84+ SE for an 89t at the beginning of this year and i love it. the CAS is a really nice feature. I would have to agree that the BASIC programing isn't as good as it was on the 84. however if you are interested in trying C, you could check out tigcc. there is also an oncalc compiler available and it works well in my opinion, though it takes a long time to write anything. I think its a great calculator, definitely improved my math grade.