ArcTan[x, y] == ArcTan[y/x] /; Element[y, Reals] && Inequality[-(Pi/2), Less, Arg[x], LessEqual, Pi/2]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], "\[Equal]", RowBox[List["ArcTan", "[", FractionBox["y", "x"], "]"]]]], "/;", RowBox[List[RowBox[List["y", "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "x", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> y </mi> <mi> x </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &lt; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> <apply> <arctan /> <apply> <times /> <ci> y </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> y </ci> <reals /> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <lt /> <apply> <arg /> <ci> x </ci> </apply> <leq /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", RowBox[List["x_", ",", "y_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcTan", "[", FractionBox["y", "x"], "]"]], "/;", RowBox[List[RowBox[List["y", "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "x", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]]]

2001-10-29