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Math Help - Verifying Trig Identity

  1. #1
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    Verifying Trig Identity

    I have one more that has me stumped. In verifying the following identity,
    I came up with a form that was close to the desired form, but ........
    Well you know what they say about getting close. I rechecked by algebra and it all looks good. Help!

    sinh 3x = 3sinh x + 4 (sinh^3) x

    NOTE: In case its not clear the last term is 4 times sinh cubed x, where the sinh function is cubed.
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  2. #2
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    Quote Originally Posted by kaiser0792 View Post
    I have one more that has me stumped. In verifying the following identity,
    I came up with a form that was close to the desired form, but ........
    Well you know what they say about getting close. I rechecked by algebra and it all looks good. Help!

    sinh 3x = 3sinh x + 4 (sinh^3) x

    NOTE: In case its not clear the last term is 4 times sinh cubed x, where the sinh function is cubed.
    These are not trigonometric functions, they're actually hyperbolic functions.

    But never mind...

    \sinh{3x} = \sinh{(2x + x)}


    Use the identity \sinh{(a + b)} = \sinh{a}\cosh{b} + \cosh{a}\sinh{b}


    \sinh{(2x + x)} = \sinh{2x}\cosh{x} + \cosh{2x}\sinh{x}

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

     = 2\sinh{x}\cosh^2{x} + 2\sinh^3{x} + \sinh{x}

     = 2\sinh{x}(1 + \sinh^2{x}) + 2\sinh^3{x} + \sinh{x}

     = 2\sinh{x} + 2\sinh^3{x} + 2\sinh^3{x} + \sinh{x}

     = 3\sinh{x} + 4\sinh^3{x}.
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  3. #3
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     3 \ \sinh x + 4 \ \sinh^{3} x

     = 3 \ \frac{e^{x}-e^{-x}}{2} + 4 \ \frac{(e^x-e^{-x})^{3}}{8}

     = 3 \ \frac{e^{x}-e^{-x}}{2} +  \ \frac{e^{3x}-3e^{x}+3^{-x}-e^{-3x}}{2}

     = \frac{e^{3x}-e^{-3x}}{2} = \sinh 3x
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