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 ArcCos

 http://functions.wolfram.com/01.13.16.0173.01

 Input Form

 ArcCos[x] - ArcCot[y] == Pi/2 - ((Sqrt[1 - x^2] + x y) (Pi - 2 ArcSin[(Sqrt[1 - x^2] - x/y)/Sqrt[1 + 1/y^2]]) + 2 Pi (Sqrt[1 - x^2] + x y + Sqrt[1 + 1/y^2] y Sqrt[(1 + 2 x Sqrt[1 - x^2] y + x^2 (-1 + y^2))/(1 + y^2)]) Floor[(Arg[I x + Sqrt[1 - x^2]] + Arg[(I - 1/y)/Sqrt[1 + 1/y^2]])/ (2 Pi)] + 2 Pi (Sqrt[1 - x^2] + x y - Sqrt[1 + 1/y^2] y Sqrt[(1 + 2 x Sqrt[1 - x^2] y + x^2 (-1 + y^2))/(1 + y^2)]) Floor[-((-Pi + Arg[I x + Sqrt[1 - x^2]] + Arg[(I - 1/y)/Sqrt[1 + 1/y^2]])/(2 Pi))])/ (2 Sqrt[1 + 1/y^2] y Sqrt[(Sqrt[1 - x^2] + x y)^2/(1 + y^2)])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCos", "[", "x", "]"]], "-", RowBox[List["ArcCot", "[", "y", "]"]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "-", FractionBox["x", "y"]]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]], "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]], ")"]]]]]]]]]]

 MathML Form

 cos - 1 ( x ) - cot - 1 ( y ) π 2 - 1 2 1 + 1 y 2 y ( x y + 1 - x 2 ) 2 y 2 + 1 ( ( x y + 1 - x 2 ) ( π - 2 sin - 1 ( 1 - x 2 - x y 1 + 1 y 2 ) ) + 2 π ( x y + ( y 2 - 1 ) x 2 + 2 1 - x 2 y x + 1 y 2 + 1 1 + 1 y 2 y + 1 - x 2 ) arg ( x + 1 - x 2 ) + arg ( - 1 y 1 + 1 y 2 ) 2 π + 2 π ( x y - 1 + 1 y 2 ( y 2 - 1 ) x 2 + 2 1 - x 2 y x + 1 y 2 + 1 y + 1 - x 2 ) - arg ( x + 1 - x 2 ) + arg ( - 1 y 1 + 1 y 2 ) - π 2 π ) x -1 y 2 -1 -1 1 2 1 1 y 2 -1 1 2 y x y 1 -1 x 2 1 2 2 y 2 1 -1 1 2 -1 x y 1 -1 x 2 1 2 -1 2 1 -1 x 2 1 2 -1 x y -1 1 1 y 2 -1 1 2 -1 2 x y y 2 -1 x 2 2 1 -1 x 2 1 2 y x 1 y 2 1 -1 1 2 1 1 y 2 -1 1 2 y 1 -1 x 2 1 2 x 1 -1 x 2 1 2 -1 1 y -1 1 1 y 2 -1 1 2 -1 2 -1 2 x y -1 1 1 y 2 -1 1 2 y 2 -1 x 2 2 1 -1 x 2 1 2 y x 1 y 2 1 -1 1 2 y 1 -1 x 2 1 2 -1 x 1 -1 x 2 1 2 -1 1 y -1 1 1 y 2 -1 1 2 -1 -1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCos", "[", "x_", "]"]], "-", RowBox[List["ArcCot", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "-", FractionBox["x", "y"]]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]], "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], "+", RowBox[List["x", " ", "y"]]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02

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