Trig. Identity Problems?

Question

How do I prove the following identities?:

(a) sin(u)sin(v)= 1/2(cos(u-v) -cos(u+v))
(b) cos(u)cos(v)= 1/2(cos(u-v) +cos(u+v))
(c) sin(u)cos(v)= 1/2(sin(u-v) +sin(u+v)

Thanks.

Answer 1

u need to have knowledge of vectors for this

Answer 2

I only have time to do the 1st. Should be enough to give you the idea.

1/2[ cos (u-v) - cos(u+v)] =
1/2[ cos u cos v + sin u sin v - ( cos u cos v - sin u sin v)] =
1/2[ cos u cos v + sin u sin v - cos u cos v + sin u sin v)] =
1/2[ sin u sin v + sin u sin v] =
sin u sin v

start with the right side, use sum and difference rules, watch your signs, simplify.

Answer 3

I will do the first and you the others.

For each one you must use the formulas:
cos(a+b) = cosacosb - sinasinb
cos(a-b) = cosacosb + sinasinb
sin(a+b) = sinacosb + sinbcosa
sin(a-b) = sinacosb - sinbcosa

a) you develop the second member only;
1/2(cos(u-v) - cos(u+v)) = 1/2(cosucosv + sinusinv -(cosucosv -sinusinv)) = 1/2( 2sinusinv) = sinusinv... you got the first member.

Go on... if you understook... you are able to finish the task.