Well, white space for one thing. The product r*Cos[t], which may be written r Cos[t], is definitely not the same thing as rCos[t] (notice lack of a space).

Then there is the problem that you can only eliminate one variable if given two equations. In attempting to eliminate two of them, Eliminate might not select the one you really want.

This may be more to your liking.

`eqn = {x == r Cos[t], y == r Sin[t]}
Eliminate[eqn, {t}]

Out[695]= {x == r Cos[t], y == r Sin[t]}

During evaluation of In[695]:= Eliminate::ifun: Inverse functions are being used by Eliminate, so some solutions may not be found; use Reduce for complete solution information. >>

Out[696]= r^2 - y^2 == x^2`