Prove that sums of the form

are equal to sums of the form

*Proof.* We compute this directly, substituting in the formulas for sine and cosine in terms of the complex exponential,

that we derived in this exercise. So, we have

where

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Stumbling Robot

A Fraction of a Dot
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Prove that sums of trig functions can be expressed as sums of complex exponentials

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Prove that sums of the form

are equal to sums of the form

*Proof.* We compute this directly, substituting in the formulas for sine and cosine in terms of the complex exponential,

that we derived in this exercise. So, we have

where

Thanks for the solution! Very small typo: c(0) is a_0/2

Thanks for the solution. Tiny typo it is c(0) = 1/2*a(0).

Hi there! Very nice proof! With very minor typo as should be .