Question

Show that cos((3x)/2)-cos(x/2)=-2sinxsin(x/2). See the attached image for more clarification. Any ideas?

Answer 1

^I'm not sure how to get from the first equation to the second one.

Answer 2

There is a general form
cosC-cosD=-2sin(C+D)/2sin(C-D)/2
u hav to use it.

Answer 3

here C=3theta/2 and D=theta/2

Answer 4

Right, factor formulae, forgot about that. Thanks!

Answer 5

yw:) Can u continue after that? if u need any help plz ask:)

Answer 6

Yep, I've solved the problem now, thanks. Just need more practice recognizing patterns like that.

Answer 7

\[\cos(\frac{3\phi}{2})-\cos(\frac{\phi}{2})=-2\sin(\theta)\sin(\frac{\theta}{2})\]
\[\frac{e^{i\frac{3}{2}\theta}+e^{-i\frac{3}{2}\theta}}{2}-(\frac{e^{i\frac{1}{2}\theta}+e^{-i\frac{1}{2}\theta}}{2})=-2(\frac{e^{i\theta}-e^{-i\theta}}{2i})(\frac{e^{\frac{i\theta}{2}}-e^{\frac{-i\theta}{2}}}{2i})\]
working with the rightside:
\[\frac{-2}{4(-1)}({e^{i\theta}-e^{-i\theta}})({e^{\frac{i\theta}{2}}-e^{\frac{-i\theta}{2}}})\]
\[\frac{-1}{2}(e^{i\theta(1+\frac{1}{2})}-e^{i\theta(1-\frac{1}{2})}-e^{-i\theta(1-\frac{1}{2})}+e^{-i\theta(1+\frac{1}{2})})\]
\[(\frac{e^{i\frac{3}{2}\theta}+e^{-i\frac{3}{2}\theta}}{2}-\frac{e^{i\frac{1}{2}\theta}+e^{-i\frac{1}{2}\theta}}{2})\]
which is your left hand side

Answer 8

Woah, thank you for showing that! Good to see how it all works.