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### 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/
```
Associated Topics:
College Trigonometry
High School Trigonometry

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