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Something that FullSimplify knows but Reduce does not ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg118014] Something that FullSimplify knows but Reduce does not ?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 8 Apr 2011 04:16:00 -0400 (EDT)

Consider the following (parametric) function of t:

g[t_] := (1/2)*Sin[2*t]*(((a - b)*(a + b))/
          Sqrt[a^2*Sin[t]^2 + b^2*Cos[t]^2] +
        (b^2 - a^2)/Sqrt[a^2*Cos[t]^2 + b^2*Sin[t]^2])

We try to find the roots of this for 0<=t<=Pi/4 using Reduce:

 Reduce[
 g[t] == 0 && a > 0 && b > a && 0 <= t <= Pi/4, t, Reals]

Reduce[
 g[t] == 0 && a > 0 && b > a && 0 <= t <= Pi/4, t, Reals]

During evaluation of In[474]:= Reduce::ztest:Unable to decide whether numeric quantities HoldForm[{Pi - 8*ArcTan[Sqrt[3 - 2*Sqrt[2]]], -Pi + 8*ArcTan[Sqrt[3 - 2*Sqrt[2]]]}] are equal to zero. Assuming they are. >>


a > 0 && b > a && (t == 2*ArcTan[Sqrt[3 - 2*Sqrt[2]]] || t == 0)

So, Reduce claims that it was unable to decide if Pi - 8*ArcTan[Sqrt[3 - 2*Sqrt[2]]] is zero or not. But

FullSimplify[Pi - 8*ArcTan[Sqrt[3 - 2*Sqrt[2]]]]

0

Is everything here really as it should be?


Andrzej Kozlowski


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