Proving an identity involving hyperbolic functions

[tamtam402]

Homework Statement

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

Homework Equations

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

= [ e^ixe^y - 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

 

[HallsofIvy]

Homework Statement

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

Homework Equations

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

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

e^{i(x- iy}= e^{ix+ y}= e^ixe^y and
e^{-i(x- iy)}= e^{-ix- y}= e^-ixe^-y
What you can do is "add and subtract the same thing":
e^ixe^y- e^-ixe^y+ e^-ixe^y+ e^-ixe^-y
= (e^ix- e^-ix)e^y+ e^-ix(e^y- e^-y)

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

= [ e^ixe^y - 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

 

[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.