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$$\frac{e^{j \omega } + e^{-j \omega }}{2}$$

##sin(\omega)## is

$$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$

I also know that ##cos(\omega - \pi / 2) = sin(\omega)##.

I've been trying to show this using exponentials, but I can't seem to manipulate one form into the other. Specifically, I can't figure out how to manipulate

$$\frac{e^{j (\omega - \pi / 2)} + e^{-j( \omega - \pi / 2)}}{2}$$

into

$$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$

or the other way around.

I seem to remember from back in school that there was some trick to this, but I can't remember it. I keep getting stuck on the fact that one has a plus between the exponential terms, and one has a minus.