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Re: Showing that ArcSinh[2]/ArcCsch[2] is 3?

• To: mathgroup at smc.vnet.net
• Subject: [mg73520] Re: Showing that ArcSinh[2]/ArcCsch[2] is 3?
• From: "Brian Gladman" <brg at nowhere.org>
• Date: Wed, 21 Feb 2007 01:41:16 -0500 (EST)
• References: <erelp9$7mm$1@smc.vnet.net>

"David W.Cantrell" <DWCantrell at sigmaxi.net> wrote in message 
news:erelp9$7mm$1 at smc.vnet.net...
>I hope I've just overlooked something very simple.
> I want to transform ArcSinh[2]/ArcCsch[2] to 3, using just "knowledge"
> already implemented in Mathematica. I tried FullSimplify first, and it
> doesn't help. I tried several other things. For example,
>
> TrigToExp[ArcSinh[2]/ArcCsch[2]]  yields
>
> Log[2 + Sqrt[5]]/Log[1/2 + Sqrt[5]/2].
>
> But then how should we transform that to 3?

I imagine that you would consider:

Log[2 + Sqrt[5]]/(Log[Expand[((1 + Sqrt[5])/2)^3]]/3)

to be cheating :-)

    Brian Gladman