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I give Mathematica these commands:

$Assumptions = Element[x, Reals] && Element[y, Reals]
\[Theta] = ArcTan[x, y]
z = Exp[I \[Theta]]
Conjugate[z]

I get

E^(-I Conjugate[ArcTan[x, y]])

How do I get Mathematica to recognize that ArcTan[x,y] is real?

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Use ComplexExpand and Simplify, Note that $Assumptions is only used by functions that use the option Assumptions

$Assumptions = Element[x, Reals] && Element[y, Reals];
θ = ArcTan[x, y];
z = Exp[I θ];
Conjugate[z] // 
  ComplexExpand[#, TargetFunctions -> {Re, Im}] & // Simplify

(* (x - I y)/Sqrt[x^2 + y^2] *)
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