There is no a "exact definition" polynomial as answer to 3(sin(x))^4+(cos(x)^)=0
my AFX ROM 1.05 gives me (pretty fast):
tan(x/2)^8 - 4tan(x/2)^6...... a loong result. i think that you can't solve it by hand easy, and you cant put it in the polynomial solver of the calc. The only way to solve it is using numeric solver.. and depending of the initial root guess, you can receive each different root of this "poly". Or using LNA with a metod that find all the roots on the given interval. in summary there isn't a result of the given eqn, is only a simplyfied one.
Classpad 300 O.S 2.20 gives me.. no answer

. Same as CP Manager v2.0
I can understand whyCP can't solve this. The cas would be programmed for solutions only (x=-/-++-%*^&), no simplifications, remember that CP is a didactic focused device.
Good find Vanhoa, go ahead man!. Comparing AFX vs CP is a hard task because there is no absolute superiority on a side. The CP is stronger than AFX in things such as integrals and some limits. On numerical matrix algebra , at least O.S 2.2, AFX beats classpad, (that is if you don't use Lua,CPLua beats any calc on the world

) . But AFX CAS runs slower things as Eigenvalues and Eigenvectors analytically.
Summarized:
DiffEqns ---> AFX
Numeric Solve ----> AFX-CP . needs some serious benchmarks. CP advantages, Natural Disp and large variable names.
Poly Solve ---> I think that AFX culd be faster, but i'm not sure.
Matrix numeric: AFX
Matrix algebraic: CP of course
if you can do some benchmarking Vanhoa on all the subjects, we're very pleased to you