Expert: Steve Holleran - 3/14/2008

Question
prove cosu-cosv=-2sin(u+v/2)sin(u+v/2)

Answer
Hi Ron,

I think there's a typo here, so I'm going to write and work with what I think you meant to write.  I think on the right side, you want the last expression to be "sin(u-v/2)", no? Because the way its written, the right side becomes "-2sin^2 (u+v/2)".

Let me work on what I think you want to prove, which is:

 cos u - cos v = -2 sin(u+v/2) sin(u-v/2).

First, let's make a substitution:

Let a = u+v/2 and b = u-v/2.

Then a+b = u+v/2 + u-v/2 = 2u/2 = u

    a-b = u+v/2 - u-v/2 = 2v/2 = v

Okay, then the left side of your expression becomes

 cos u - cos v = cos (a+b) - cos (a-b).

Using the addition and subtraction laws for cosine,

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

           = cos a cos b - sin a sin b - cos a cos b - sin a sin b

           = -2 * sin a sin b

           = -2 * sin(u+v/2) sin (u-v/2)

I hope this is what you meant.  

Steve