Prove that for positive integers *k*, *m*, and *n*, that

.

First, we distribute the into the sum:

;

next, we use the product identity for sines:

, which means for the terms in the sum, we have

.

Now, notice that this sum is

, and thus is a telescoping series:

,

and we can use our product rule for sines again, in the reverse direction; with , and , we see

,

and we have our proof.

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Tags: Math, Monday Math, Telescoping Series, Trigonometry

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