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)$ .