clicliclic@freenet.de schrieb:   
   >   
   > Albert Rich schrieb:   
   > >   
   > > On Wednesday, October 4, 2017 at 6:21:59 AM UTC-10, clicl...@freenet.de   
   wrote:   
   > >   
   > > > I believe that:   
   > > >   
   > > >   SQRT(x^2/(1 + SQRT(x^2 - 1))^2)*(1 + SQRT(x^2 - 1))/x = x/SQRT(x^2)   
   > > >   
   > > > FriCAS will probably squeal, but can Mathematica or Maple validate   
   > > > or invalidate this simplification?   
   > > >   
   > >   
   > > Dividing both sides of the proposed identity by (1+sqrt(x^2-1))/x   
   > > gives   
   > >   
   > >   sqrt(x^2/(1+sqrt(x^2-1))^2) = sqrt(x^2)/(1+SQRT(x^2-1))   
   > >   
   > > which is an equation of the form sqrt(z^2)=z with   
   > > z=sqrt(x^2)/(1+SQRT(x^2-1)).   
   > >   
   > > sqrt(z^2)=z is valid iff re(z)>=0 for all complex z. Thus to prove   
   > > the proposed identity need to show re(sqrt(x^2)/(1+SQRT(x^2-1)))>=0   
   > > for all complex x. Any ideas?   
   > >   
   >   
   > I think your equivalence must be amended as follows:   
   >   
   >   sqrt(z^2) = z iff re(z) > 0 or (re(z) = 0 and im(z) >= 0)   
   >   
   > And for simplicity one may consider:   
   >   
   >   z = re(sqrt(w)/(1+sqrt(w-1)))   
   >   
   > for all complex numbers w = x^2.   
   >   
      
   There doesn't seem to be much interest in this, so here's my sketch of   
   a proof. By use of:   
      
     RE(a/b) >= 0 IFF RE(a*CONJ(b)) = RE(a)*RE(b) + IM(a)*IM(b) >= 0   
      
   the proof of RE(z) >= 0 becomes straightforward:   
      
     RE(SQRT(w)/(1+SQRT(w-1))) >= 0   
      
     RE(SQRT(w))*RE(1+SQRT(w-1)) + IM(SQRT(w))*IM(SQRT(w-1)) >= 0   
      
   which holds because of:   
      
     RE(SQRT(w)) >= 0, RE(1+SQRT(w-1)) >= 1,   
      
     IM(SQRT(w)) >= 0 IFF IM(SQRT(w-1)) >= 0   
      
   in consideration of the complex phases with special attention paid to   
   0 < w < 1.   
      
   It remains to establish that RE(z) = 0 AND IM(z) < 0 never happens.   
   RE(z) = 0 requires RE(SQRT(w)) = 0, hence w <= 0, which in turn implies   
   RE(z) = 0 only for IM(SQRT(w)) = 0, hence w = 0. So there is no other   
   candidate but z = SQRT(0)/(1+SQRT(0-1)) = 0, with IM(z) = 0   
   contradicting IM(z) < 0. Q.E.D.   
      
   Martin.   
      
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