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Proving an identity involving hyperbolic functions

  1. May 30, 2012 #1
    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. jcsd
  3. May 30, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    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.

     
  4. May 30, 2012 #3
    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.
     
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