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.
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