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Thread: Proof of inverse cosech(x)

  1. January 26th 2010, 12:42 PM #1
    JQ2009
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    Proof of inverse cosech(x)

    How would you prove cosech^-1(x) = sinh^-1(-1/x)?

    Both cosech and sinh are meant to mean inverse. Thanks in advance.
    Last edited by JQ2009; January 26th 2010 at 01:06 PM.
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  2. January 26th 2010, 01:29 PM #2
    Jhevon
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    Quote Originally Posted by JQ2009 View Post
    How would you prove cosech^-1(x) = sinh^-1(-1/x)?

    Both cosech and sinh are meant to mean inverse. Thanks in advance.
    let \sinh^{-1} \frac 1x = y. solve for x and find the \text{csch}^{-1} x from that

    Spoiler:

    \sinh^{-1} \frac 1x = y

    \Rightarrow \frac 1x = \sinh y

    \Rightarrow x = \frac 1{\sinh y} = \text{csch } y

    \Rightarrow \text{csch}^{-1} x = y = \sinh^{-1} \frac 1x
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