# Hyperbolic cosine

1. Jul 23, 2007

### bakin

1. The problem statement, all variables and given/known data
Prove the identity:

Cosh(x) + Cosh(y) = 2Cosh[(x+y)/2]Cosh[(x-y)/2]

2. Relevant equations
Cosine sum-to-product
http://library.thinkquest.org/17119/media/3_507.gif

3. The attempt at a solution
Can you use the same formula for Cosine sum to product for hyperbolic cosine?

Thanks!

Last edited by a moderator: Apr 22, 2017
2. Jul 23, 2007

### robphy

Do you know how cos x is related to cosh x?
Or more precisely, how x and y are related if cos x=cosh y?

To prove your identity, you could use a variant of the proof you hopefully used for cos.

3. Jul 23, 2007

### bakin

I'm not sure how they're related, we went through them very quickly and very briefly. I do know that cosh(x) is e^x + e^-x all over 2, but we didn't spend a lot of time on them.

4. Jul 23, 2007

### Dick

The relation is pretty simple, cosh(x)=cos(i*x). It's pretty easy to show this using the power series for cos(x) and e^x. Does that help?

5. Jul 25, 2007

### Gib Z

Or Alternatively, from Euler's Identity a definition of cosine follows:
$$\cos x = \frac{ e^{ix} - e^{-ix}}{2}$$. That explains the connection.

As for another method to prove the relation, replace all the Hyperbolic Cosines with their exponential definition and rearrange into what you want to see.

6. Jul 25, 2007

### bakin

That works! I replaced the right side into the definition, combined terms, and then separated, and got it. thanks a lot guys.