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Missing simplification for ArcSin

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  • Subject: [mg127945] Missing simplification for ArcSin
  • From: Roland Franzius <roland.franzius at>
  • Date: Tue, 4 Sep 2012 05:47:06 -0400 (EDT)
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While working on the solution of the free motion on a sphere one 
encounters the energy integral

{d\[Tau]->(2 du^2/4)/Sqrt[(1+\[Mu])u^2- (1+u^2/4)^2]}   ,   \[Mu]>0

in the plane of the stereographic map

u = 2 ArcTan[theta/2]

Of course one expects a solution in ArcSin form, since its only a tilted 
motion with constant speed on the equator, but Mathematica delivers a 
complex ArcTan wich is very difficult to Reduce to Reals.

At this I found the all time not detected trig flaw in Mathematica:

In: FullSimplify[{ArcTan[x] - ArcSin[x/Sqrt[1 + x^2]] == 0}, Trig -> True,
   Assumptions -> x > 0]

Out: {ArcSin[x/Sqrt[1 + x^2]] == ArcTan[x]}

does not work. This may be traced to the incapability to work on complex 
Logs of Reals

In: Assuming[0 < x < 1,
  FullSimplify[TrigToExp[{ArcTan[x] - ArcSin[x/Sqrt[1 + x^2]] }]]]

Out: {1/2 \[ImaginaryI] (Log[1 + \[ImaginaryI] x] +
     Log[-(\[ImaginaryI]/(-\[ImaginaryI] + x))])}

So, while waiting for an improvement in this 
trig-simplification-desert, I am using the standard formula set generated by

ArcFunctionsReplacements =
  Outer[FullSimplify[#2[x] -> InverseFunction[#1][#1[#2[x]]]] &, {Sin,
    Cos, Tan}, {ArcSin, ArcCos, ArcTan}]

{{ArcSin[x] -> ArcSin[x], ArcCos[x] -> ArcSin[Sqrt[1 - x^2]],
   ArcTan[x] -> ArcSin[x/Sqrt[1 + x^2]]},
{ArcSin[x] ->  ArcCos[Sqrt[1 - x^2]], ArcCos[x] -> ArcCos[x],
   ArcTan[x] -> ArcSec[Sqrt[1 + x^2]]},
{ArcSin[x] ->  ArcTan[x/Sqrt[1 - x^2]],
  ArcCos[x] -> ArcCot[x/Sqrt[1 - x^2]], ArcTan[x] -> ArcTan[x]}}


Roland Franzius

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