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Can I express a function as a series of another set of functions

Let me give an example

Suppose I want to expand $\cos(x)^{4}$ as in a series of functions $C_{k}=(1/2-\cos(kx))$. That is $\cos(4x)=k_0 C_0+k_1 C_1 + k_2 C_2 \cdots$ and I am looking for coefficients $k$

Now, I can use trig_reduce() to express $\cos(x)^{4}$ as a sum $\cos(kx)$, but how do I get get $\cos(x)^4$ in terms of $C_{k}$.