Proving Hyperbolic Identity Using Osborn's Rule

[hex.halo]

Homework Statement

Given the trigonometric identity cos(x+y)... use Osborn's rule to write down the corresponding identity for cosh(x+y)... Use the definitionis of the hyperbolic functions to prove this identity

Homework Equations

The Attempt at a Solution

I can use Osborns rule to find the hyperbolic equivilent of the identity, however, I don't understand how I am to prove this identity...

 

[rock.freak667]

$cos(x+y)=cosxcosy-sinxsiny$

Osborn said that when you have the product of two sines, you replace the sines with sinh and a negative sign.

so therefore $cosh(x+y)=cosxcosy-(-sinhxsinhy)$

 

[exk]

rock.freak missed the h in his last bit for the 2 cos terms on the rhs.
$cosh(x+y)=coshxcoshy-(-sinhxsinhy)$

 

[HallsofIvy]

Homework Statement

Given the trigonometric identity cos(x+y)... use Osborn's rule to write down the corresponding identity for cosh(x+y)... Use the definitionis of the hyperbolic functions to prove this identity

Homework Equations

The Attempt at a Solution

I can use Osborns rule to find the hyperbolic equivilent of the identity, however, I don't understand how I am to prove this identity...

Okay, you already know that cosh(x+ y)= cosh(x)cosh(y)+ sinh(x)sinh(y). Now replace cosh(x) by $(e^x+ e^{-x})/2$, replace sinh(x)= [itex](e^x- e{-x})/2[/tex], the corresponding things for cosh(y) and sinh(y) and do the algebra. What do you get when you multiply<br /> $$\frac{e^x+ e^{-x}}{2}\frac{e^y+ e^{-y}}{2}$$[/itex]