# Trig identity in complex multiplication

Just wondering how this is simplified to the third line:

If w, z are complex numbers

wz = rs( cos$$\alpha$$ + isin $$\alpha$$ ) (cos $$\varphi$$ + isin $$\varphi$$)

wz = rs(cos$$\alpha$$ cos $$\varphi$$ - sin $$\alpha$$sin$$\varphi$$) + i(sin $$\alpha$$cos$$\varphi$$ + cos $$\alpha$$ sin $$\varphi$$))

wz = rs(cos ($$\alpha$$ +$$\varphi$$) + i sin($$\alpha$$ +$$\varphi$$))

What sort of trigonometric identity is used here between the 2nd and 3rd lines?

Exactly those as written:

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

Thanks I hadn't seen these identities before

mathman
Thanks I hadn't seen these identities before
These are basic identities, which are taught in the first course of trigonometry.

You would think so but apparently my school doesnt see the importance in teaching this stuff. The only identity we were taught was

sin^2(x) + cos^2(x) = 1

not even all the half angle ones which I'm finding out about now... How helpful for me!

mathman
It looks like your school must teach trigonometry for a couple of weeks within a broader math course. When I was in high school, we had a one semester course for trig.