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

• To: mathgroup at smc.vnet.net
• Subject: [mg73512] Re: [mg73499] Showing that ArcSinh[2]/ArcCsch[2] is 3?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 21 Feb 2007 01:36:59 -0500 (EST)
• References: <200702201119.GAA07770@smc.vnet.net>

On 20 Feb 2007, at 12:19, David W.Cantrell wrote:

> 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?
>
> David
>


As is often the case with FullSimplify, it is easier to prove that  
one thing is equal to another (or, equivalently, that the difference  
between them is 0) than to transform one of them into the other. In  
fact,  Simplify and FullSimplify are not intended for "transforming"  
an expression into another specific expression, although, of course,  
they are often used for this purpose. Anyway, in this case:


FullSimplify[ArcSinh[2]/ArcCsch[2]-3]

0

or


FullSimplify[ArcSinh[2] - 3*ArcCsch[2]]

0

Andrzej Kozlowski

• References:
  • Showing that ArcSinh[2]/ArcCsch[2] is 3?
    • From: "David W.Cantrell" <DWCantrell@sigmaxi.net>