## anonymous one year ago Verify: sin (x+y)cos(x-y)=sin(x) cos(x)+cos(y)sin(y)

1. welshfella

use the compound angle formula to expand the left side then simplify. I think you will be able to make use of the identity sin^2 x + cos^2 x = 1

2. welshfella

sin(x + y) = sin x cos y + sin y cos x cos (x - y) = cos x cos y + sin x sin y

3. anonymous

Well once it's broken down where do I go from there?

4. welshfella

you should get what is on the right side so proving the identity

5. welshfella

its a bit long winded but that is the way to do it

6. anonymous

Can u show me all the way to the answer? My brain is ready to fall outta my ears from trying to figure this out.

7. zepdrix

Ehhh I can maybe show you a few steps -_- let's see here....

8. zepdrix

$\large\rm \color{orangered}{\sin(x + y) = \sin x \cos y + \sin y \cos x}$$\large\rm \color{royalblue}{\cos (x - y) = \cos x \cos y + \sin x \sin y}$We'll apply these identities to our problem:$\large\rm \color{orangered}{\sin(x+y)}\color{royalblue}{\cos(x-y)}$Which will give us:$\large\rm \color{orangered}{\left[\sin x \cos y + \sin y \cos x\right]}\color{royalblue}{\left[\cos x \cos y + \sin x \sin y\right]}$Ok with that first step? :)

9. anonymous

Yes I understand that!

10. zepdrix

So hmm.. I guess we have to FOIL from here.

11. zepdrix

$\rm =\color{orangered}{\sin x \cos y}\color{royalblue}{\cos x \cos y}+\color{orangered}{\sin x \cos y}\color{royalblue}{\sin x \sin y}\\+\color{orangered}{\sin y \cos x}\color{royalblue}{\cos x \cos y}+\color{orangered}{\sin y \cos x}\color{royalblue}{\sin x \sin y}$Which becomes this I suppose.