Prove Hyperbolic Cosine Sum-to-Product Identity

[bakin]

Homework Statement

Prove the identity:

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

Homework Equations

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

The Attempt at a Solution

Can you use the same formula for Cosine sum to product for hyperbolic cosine?

Thanks!

 

[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.

 

[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.

 

[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?

 

[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.

 

[bakin]

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