Function FullSimplify unable to simplify relatively simple algebraic expression

Question

I want to simplify with Mathematica 11 the following algebraic expression under the assumptions $x<0$ and $y>0$:

A= (-x + Sqrt[x^2 + y])/Sqrt[2 x^2 + y - 2 x Sqrt[x^2 + y]]

A short calculation yields A = 1. But Mathematica seems not able to obtain that result when the following expression is evaluated:

FullSimplify[A,Assumptions->x<0&&y>0]

Is it possible to help Mathematica to solve this?

Thanks for your answer.

Answer 1

One possibility is to add Reduce to the list of TransformationFunction rules. For example:

oneReduce[x_] := If[TrueQ @ !Reduce[{x != 1, $Assumptions}], 1, x]

If Reduce can determine that x != 1 is definitely False based on $Assumptions, then it transforms x to 1. Let's see this in action:

Assuming[x<0 && y>0, Simplify[A, TransformationFunctions->{Automatic, oneReduce}]]

1

Another useful transformation function is:

zeroReduce[x_] := If[TrueQ @ !Reduce[{x != 0, $Assumptions}], 0, x]

Perhaps this approach will work with more complicated expressions.