ArcTan[x] - ArcTan[y] == ArcTan[(x - y)/(1 + x y)] + (Pi/2) (1 - Sign[x y + 1]) Sign[x] /; x y != -1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "x", "]"]], "-", RowBox[List["ArcTan", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List["x", "-", "y"]], RowBox[List["1", "+", RowBox[List["x", " ", "y"]]]]], "]"]], "+", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["Sign", "[", "x", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["x", " ", "y"]], "\[NotEqual]", RowBox[List["-", "1"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mi> y </mi> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> &#8800; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arctan /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Sign </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <neq /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTan", "[", "x_", "]"]], "-", RowBox[List["ArcTan", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List["x", "-", "y"]], RowBox[List["1", "+", RowBox[List["x", " ", "y"]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["Sign", "[", "x", "]"]]]]]], "/;", RowBox[List[RowBox[List["x", " ", "y"]], "\[NotEqual]", RowBox[List["-", "1"]]]]]]]]]]

2001-10-29