I'm trying to coax Mathematica into rewriting $ x\sqrt{1-y^2/x^2} = \text{sgn}(x)\sqrt{x^2-y^2} $ for $ x $ and $ y $ real. I have tried all combinations of FullSimplify
and Refine
with the appropriate assumptions. Mathematica refuses to cooperate.
With the additional assumption $ x>0 $, I can get Mathematica to simplify to $ \sqrt{x^2-y^2} $, but I don't want to assume this. I want the general result with the $ \text{sgn}(x) $.
*Simplify
etc. refuse to carry out further transformations, that means, Mathematica determines, according to its own built-in principles, that the "simplest" form has already been reached. $\endgroup$ – Αλέξανδρος Ζεγγ Nov 18 '18 at 14:13