Re: How to simplify ArcCos[x/Sqrt[x^2+y^2]] to Pi/2-ArcTan[x/Abs[y]]?

• To: mathgroup at smc.vnet.net
• Subject: [mg91834] Re: How to simplify ArcCos[x/Sqrt[x^2+y^2]] to Pi/2-ArcTan[x/Abs[y]]?
• From: Roland Franzius <roland.franzius at uos.de>
• Date: Tue, 9 Sep 2008 06:59:03 -0400 (EDT)
• References: <ga2po2\$p77\$1@smc.vnet.net>

```Peng Yu schrieb:
> Hi,
>
> ArcCos[x/Sqrt[x^2+y^2]]
>
> and
>
> Pi/2-ArcTan[x/Abs[y]]
>
> are the same.
>
> But I can not get the first expression be simplified to the second
> one. And the following command can not be simplified to zero in
> mathematica as well.
>
> FullSimplify[ArcCos[x/Sqrt[x^2 + y^2]] - ( Pi/2 - ArcTan[x/Abs[y]]),
>  Element[x, Reals] && Element[y, Reals]]
>
> I'm wondering if there is any walkaround to do this simplification.

Try

Assuming[x > 0 && y > 0,
FullSimplify[
TrigToExp[ArcCos[x/Sqrt[x^2 + y^2]] - (Pi/2 - ArcTan[x/Abs[y]])]]]

(1/2)*I*(Log[1/(I*x + y)] + Log[I*x + y])

Obviously Mathematica isn't making a guess to which leaf of the complex
Log your algebraic complex travel history brought even if you arrived on
the positive real line.

--

Roland Franzius

```

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