In[]:=
ClearAll["Global`*"]
(*" Ulrich Mutze 2024-07-06 There is a formula for Arg[x+y] which avoids the deficiencies ofthe formula ArcTan[y/x]. This formula, expressed in WL, isArg[x+y] == If[y!=0, 2 ArcTan[y/(x+Sqrt[x²+y²])],If[x>=0, 0, Pi]]assuming x,y ∈ Reals. I did not find it in WolframMathWorld. Anyways, it is a direct consequence of 'inscribed angles' being a half of their 'central angles' (see e.g. WolframMathWorld). "*)
In[]:=
FullSimplify[Arg[x+y]-2ArcTan[y/(x+Sqrt[x²+y²])],x∈Reals&&y∈Reals&&y!=0]
Out[]=
-2ArcTan+Arg[x+y]
y
x+
x²+y²
(*" On my system : Mathematica 13.1, Ubuntu, I the above expressiongets returned unevaluated, although the result is simply 0. How this can happen? Can it be, thatthe result, which is nearly trivially obtained by elementary geometry,nevertheless is so hard to obtain by manipulating expressions thatMathematica can't find its way through? Is there a way in WL togive a hint, e.g. by applying some sort of expansion or replacementthat the evaluation becomes possible ?"*)