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Simplifying an expression - with my own definition of what "simple"


I am using Mathematica 5.2 on i386 Linux and I have the following problem: I want to express a rational polynomial of trigonometric functions in terms of elementary polynomials that I have defined myself.

A simplified example of what I want to do is illustrated as follows: Suppose the elementary function is f[x_]:=Sin[x/2], then I want to obtain

Cos[x] becomes 1-2*f[x]^2

Sin[x]^2 becomes 4f[x]^2-4f[x]^4

and so on. So my own definition of "simple" is "in terms of f[x]".

Is there a straightforward way to do this?

My actual problem is a little harder since it involved multiple variables and rational trigonometric polynomials, but at least analytically I can show that it can always be rewritten in terms of the elementary functions. The question is if Mathematica can in principle solve such types of problems as illustrated above.

Thanks in advance!

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