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

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