I need to prove that e^(it) + e^(is) = 2cos((t-s)/2)e^(i(s+t)/2). I have used lns and gotten it to -ts = i((s+t)/2)ln(2cos((t-s)/2), but I don't know where to go from there.

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- August 28th 2007, 01:06 PMdnlstffrdProving something about complex
I need to prove that e^(it) + e^(is) = 2cos((t-s)/2)e^(i(s+t)/2). I have used lns and gotten it to -ts = i((s+t)/2)ln(2cos((t-s)/2), but I don't know where to go from there.

- August 28th 2007, 01:25 PMJhevon
It's not that bad. Here's one way to do it. You need to know three things:

**Euler's Formula:**

**Product-to-sum formulas for sine and cosine:**

and finally,

Now, let's get to it:

...........................applied Euler's formula

....................................rearranged the terms

.........applied the sum-to-product formulas

...............................factored out the common

as desired - August 28th 2007, 01:38 PMdnlstffrd
I totally forgot about Euler's formula, once I remembered that it was a cinch lol

- August 28th 2007, 01:43 PMThePerfectHacker
I have an geometric approach to this. Try to see if you can come up with it.