Sin[I ArcCot[z]] == ((1/2) I (((-I + z)/z)^I - ((I + z)/z)^I))/ (1 + 1/z^2)^(I/2)

Cell[BoxData[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", "z"]], "z"], ")"]], "\[ImaginaryI]"], "-", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], "z"], ")"]], "\[ImaginaryI]"]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mi> &#8520; </mi> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mi> &#8520; </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sin /> <apply> <times /> <imaginaryi /> <apply> <arccot /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <imaginaryi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <imaginaryi /> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", "z"]], "z"], ")"]], "\[ImaginaryI]"], "-", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], "z"], ")"]], "\[ImaginaryI]"]]], ")"]]]]]]]]

2007-05-02