Sin[n (ArcTan[z]/2)] == (z/(Sqrt[2] Sqrt[z^2])) Sqrt[1 - 1/Sqrt[z^2 + 1]] ChebyshevU[n - 1, (1/Sqrt[2]) Sqrt[1 + 1/Sqrt[1 + z^2]]] /; Element[n, Integers] && n > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["n", FractionBox[RowBox[List["ArcTan", "[", "z", "]"]], "2"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["z", " "]], RowBox[List[SqrtBox["2"], " ", SqrtBox[SuperscriptBox["z", "2"]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]]]], RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List[FractionBox["1", SqrtBox["2"]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sin /> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> ChebyshevU </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", "n_", " ", RowBox[List["ArcTan", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", FractionBox[SqrtBox[RowBox[List["1", "+", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]]], SqrtBox["2"]]]], "]"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[SuperscriptBox["z", "2"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

2001-10-29