From: peter.luschny@gmail.com   
      
   Am Mittwoch, 4. Oktober 2017 18:38:49 UTC+2 schrieb Nasser M. Abbasi:   
   > Mathematica says, for real x, the above is valid only for x<=1 || x>=1   
   > In[1]:= expr1 = Sqrt[x^2/(1 + Sqrt[x^2 - 1])^2]*((1 + Sqrt[x^2 - 1])/x);   
   > In[2]:= expr2 = x/Sqrt[x^2];   
   > In[5]:= Reduce[expr1 - expr2 == 0, x, Reals]   
   > Out[5]= x <= -1 || x >= 1   
      
   This depends on the method of verification.   
      
   Alternatively:   
      
   Simplify[Sqrt[x^2/(1 + Sqrt[x^2 - 1])^2]*((1 + Sqrt[x^2 - 1])/x)]   
   gives: 1/(sgn(x)) assuming x is real   
      
   Simplify[x/Sqrt[x^2]]   
   gives: sgn(x) assuming is real   
      
   Reduce[sgn(x) - 1/(sgn(x)) == 0, x, Reals]   
   gives: Solution over the reals: x < 0 and x > 0.   
      
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