Can the sum of square cosines be expressed as a single function of the sum of x?

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The discussion centers on the challenge of expressing the finite sum of square cosines, Σ^N_{s=1}cos²(x_s), as a single function of the sum of x, f(Σ^N_{s=1}x_s). It is noted that this transformation is not feasible, as demonstrated by examples where different pairs of x values yield the same sum but produce different results for the sum of cosines. Specifically, for N=2, varying the values of x can lead to a sum of cosines of 2 or 0, despite the sum of x remaining constant at π. The conversation suggests that while approximations like Taylor series might be considered, a direct expression is not achievable. Ultimately, the conclusion is that such a transformation cannot be performed.
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I'm looking for a way to change a finite sum of square cosines:

\Sigma^{N}_{s=1}cos^{2}(x_{s})

into a single function of the sum of x:

f(\Sigma^{N}_{s=1}x_{s})

Is there a known way to do this, even if with an approximate method (i.e. Taylor series or such)?. It's ok if it just works in a \pi range.
 
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Can't be done. If N=2, x_1=0, x_2=\pi, sum of x is \pi and sum of cosines is 2. IF N=2, x_1=\pi/2, x_2=\pi/2, sum of x is still \pi, but sum of cosines is 0.
 

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