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Math Help - Hyperbolic Identities

  1. #1
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    Hyperbolic Identities

    I cannot for the life of me work this one out, I am getting very close, but I can't take it further, I must have made a mistake somewhere down the line, anyone mind checking my work? Couldn't get a hold of my prof today

    Question: Show that \frac{sinh 3t}{sinh t}=1+2cosh 2t.

    Here's what I did.

    \frac{sinh(3t)}{sinh(t)}=1+2cosh(2t)

    \frac{sinh(2t+t)}{sinh(t)}=1+2cosh(2t)

    \frac{sinh(2t)cosh(t)+cosh(2t)sinh(t)}{sinh(t)}=1+  2cosh(2t)

    \frac{2sinh(t)cosh^2(t)+cosh(2t)sinh(t)}{sinh(t)}=  1+2cosh(2t)

    2cosh^2(t)+cosh(2t)=1+2cosh(2t)

    I've been trying all sorts of ways to simplify it from here, but none have worked.

    Thanks for your time!
    Kasper
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  2. #2
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    Quote Originally Posted by Kasper View Post
    I cannot for the life of me work this one out, I am getting very close, but I can't take it further, I must have made a mistake somewhere down the line, anyone mind checking my work? Couldn't get a hold of my prof today

    Question: Show that \frac{sinh 3t}{sinh t}=1+2cosh 2t.

    Here's what I did.

    \frac{sinh(3t)}{sinh(t)}=1+2cosh(2t)

    \frac{sinh(2t+t)}{sinh(t)}=1+2cosh(2t)

    \frac{sinh(2t)cosh(t)+cosh(2t)sinh(t)}{sinh(t)}=1+  2cosh(2t)

    \frac{2sinh(t)cosh^2(t)+cosh(2t)sinh(t)}{sinh(t)}=  1+2cosh(2t)

    2cosh^2(t)+cosh(2t)=1+2cosh(2t)

    I've been trying all sorts of ways to simplify it from here, but none have worked.

    Thanks for your time!
    Kasper
    Identity: \cosh (2t) = \cosh^2 t + \sinh^2 t = 2 \cosh^2 t - 1 \Rightarrow \cosh^2 t = \frac{\cosh (2t) + 1}{2}.
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  3. #3
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    Awesome, got it. Thanks
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