Antony F. answered • 10/24/12

The one who can get you from an F to an A+ in mathematics!

So you want to show that cosh (x + y) = cosh x cosh y + sinh x sinh y. Well be aware that

cosh x = (e^{x} + e^{-x})/2 and sinh x = (e^{x}- e^{-x})/2 for any real number x

Now, 2 **cosh (x + y) = (e ^{x+y}+ e^{-(x+y)}) = (e^{x} e^{y}+ e^{-x} e^{-y}) ...........................(A)**

4 cosh x cosh y = (e^{x} + e^{-x}) (e^{y} + e^{-y}) = (e^{x} e^{y} + e^{x} e^{-y} + e^{-x} e^{y} + e^{-x}e^{-y})

4 sinh x sinh y = (e^{x} e^{y} - e^{x} e^{-y}- e^{-x} e^{y} + e^{-x} e^{-y})

4 cosh x cosh y + 4 sinh x sinh y = 2e^{x} e^{y} + 2 e^{-x} e^{-y }= 2(e^{x+y}+ e^{-(x+y)})

Now it follows that cosh x cosh y + sinh x sinh y = (e^{x+y} + e^{-(x+y)})/2 ..............................(B)

Observe that equation (A) and (B) are identical and so

cosh (x + y) = (e^{x+y} + e^{-(x+y)})/2 = cosh x cosh y + sinh x sinh y

Walter B.

Diregard Please, I have completed the proof, I will post laster, I apologize, but this is messy.09/14/12