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I have been working on this one for a couple of hours and i just get stuck on every attempt i make.

I have to reduce the formula algebraically:

$\sinh(2 \cdot \sinh^{-1}(y))$

And I just can't seem to do it. I tried using the hyperbolic addition formulas to do something but I just ended up with an even more convoluted expression.

I tried using the addition formula with

$\sinh(x + x) = \cosh(x)\sinh(x) + \sinh(x)+\cosh(x)$

where $x$ is $\sinh^{-1}(y)$,

and then I replaced $\cosh(x)$ and $\sinh(x)$ with their definitions. It did not work.

Can anyone help me out here?

ForgotALot
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VictorVH
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1 Answers1

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Your idea of using the identity would actually work.

$\sinh(2x) = 2 \sinh(x) \cosh(x)$ where $x = \sinh^{-1}(y)$

$= 2 y \cosh(x)$.

Now $\cosh(x) = \sqrt{1 + \sinh(x)^2}$.

Can you finish?

user21820
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  • I have no clue where you are heading sorry, can you explain a bit further? – VictorVH Mar 08 '16 at 00:28
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    @VictorVH: Please tell me which line is not clear to you. – user21820 Mar 08 '16 at 00:32
  • The last two lines. I don't quite understand how you get the results and what to do with them. – VictorVH Mar 08 '16 at 00:35
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    @VictorVH: The last identity I stated comes from the basic identity $\cosh(x)^2 - \sinh(x)^2 = 1$. You use it to simplify the result you got earlier using the other identity. – user21820 Mar 08 '16 at 00:41
  • Ok, but where do i go from there? I am really not that good at simplifcation and have no idea of which direction to take it. – VictorVH Mar 08 '16 at 00:44
  • @VictorVH: Tell me what you get after combining the two results and we go from there. – user21820 Mar 08 '16 at 00:46
  • I get $2 \cdot y \cdot sqrt(1 + sinh(x)^2)$. I don't know what to do with that. Do i place the definition of x in sinh and then calculate or what do i do? – VictorVH Mar 08 '16 at 00:55
  • @VictorVH: What is $\sinh(x)$ in terms of $y$? We used this fact in the first part. Remember the definition of $x$. – user21820 Mar 08 '16 at 00:57
  • $sinh(sinh^{-1}(y)) = \frac{e^{sinh^{-1}(y)} + e^{-sinh^{-1}}{2}$ Or am i wrong? – VictorVH Mar 08 '16 at 01:01
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    What's $f(f^{-1}(y))$ for any function $f$ that has an inverse? – Lothar Narins Mar 08 '16 at 02:37
  • @VictorVH: I suspect you don't even know what you were doing in your own working in your question. Explain how I got "$2\sinh(x)\cosh(x) = 2y\cosh(x)$" in my answer, since you said you understood it. – user21820 Mar 08 '16 at 12:22