Proving Identity for Sum of Cosines

Date: 10/17/2003 at 22:01:36
From: Amanda
Subject: The Sum of Cosines

How do you prove the identity for the sum of cosines?  What are some 
good problems to test understanding of this subject?

Date: 10/18/2003 at 07:13:29
From: Doctor Luis
Subject: Re: The Sum of Cosines

Hi Amanda,

Proving the identity for sum of cosines is somewhat tricky.  Here's 
how you do it.

First, you look at the sum of two cosines

   cos(A) + cos(B)

and next, you introduce new variables x and y, defined in terms of A 
and B,

  cos(A) + cos(B) = cos(x + y) + cos(x - y)

Now, you apply the cosine addition formula to the right hand side of 
that last equation

  cos(x + y) + cos(x - y)

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

     = 2cos(x)cos(y)

We're almost done.  We just have to express x and y in terms of A and B.

  A = x + y
  B = x - y

You can see that A + B = 2x, and A - B = 2y.  This means

  x = (A + B)/2
  y = (A - B)/2

And that proves the sum of cosines formula

  cos(A) + cos(B) = 2 cos((A+B)/2) cos((A-B)/2)

To see if you understand this process, see if you can come up with 
identities for 

  sin(A) + cos(B)
  sin(A) + sin(B)
  sin(A) - sin(B)
  cos(A) - cos(B)
  sin(A) - cos(B)

Those would be good problems to do.

I hope this helped!  Let us know if you have any more questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/