# Homework Help: Proving an identity involving hyperbolic functions

1. May 30, 2012

### tamtam402

1. The problem statement, all variables and given/known data

Prove sin(x-iy) = sin(x) cosh(y) - i cos(x) sinh(y)

2. Relevant equations

3. The attempt at a solution

I tried to prove it by developing sinh into it's exponential form, but I get stuck.

sinh(x-iy) = [ ei(x-iy) - e-i(x-iy) ] /2i

= [ eixey - e-ix e-y ] /2i

This is where I get stuck. I can regroup the terms to get the following equation, but doesn't seem like the right direction.

= sinh(y+ix)/i

2. May 30, 2012

### HallsofIvy

You mean sin(x- iy) not sinh(x-iy).

ei(x- iy= eix+ y= eixey and
e-i(x- iy)= e-ix- y= e-ixe-y
What you can do is "add and subtract the same thing":
eixey- e-ixey+ e-ixey+ e-ixe-y
= (eix- e-ix)ey+ e-ix(ey- e-y)

Now convert those to sin, cos, sinh, and cosh.

3. May 30, 2012

### tamtam402

You're right, I messed up the sin -> sinh when I copied my notes.

Thanks for the tip, I knew there was a small trick I was missing. Mary L Boas(*) book is pretty good so far, but it lacks some explanations sometimes. It makes it pretty hard to rely purely on that book for self-studying :(

* What's the correct syntax to use on Boas? I know Boas's is wrong.