Cos[n ArcTan[z]] == ChebyshevT[n, 1/Sqrt[1 + z^2]]

Cell[BoxData[RowBox[List[RowBox[List["Cos", "[", RowBox[List["n", " ", RowBox[List["ArcTan", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <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> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <cos /> <apply> <times /> <ci> n </ci> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cos", "[", RowBox[List["n_", " ", RowBox[List["ArcTan", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]], "]"]]]]]]

2001-10-29