Tan[z] == z/(1 - z^2/(3 - z^2/(5 - z^2/(7 - z^2/(9 - z^2/(11 - \[Ellipsis])))))) /; !Element[z/Pi - 1/2, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Tan", "[", "z", "]"]], "\[Equal]", FractionBox["z", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["3", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["5", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["7", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["9", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["11", "-", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[FractionBox["z", "\[Pi]"], "-", FractionBox["1", "2"]]], ",", "Integers"]], "]"]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mi> z </mi> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> - </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> <mo> /; </mo> <mtext> &#8203; </mtext> <mrow> <mrow> <mfrac> <mi> z </mi> <mi> &#960; </mi> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <tan /> <ci> z </ci> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 5 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 11 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#8230; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <notin /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Tan", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["z", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["3", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["5", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["7", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["9", "-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["11", "-", "\[Ellipsis]"]]]]]]]]]]]]]]]]]], "/;", RowBox[List["!", RowBox[List[RowBox[List[FractionBox["z", "\[Pi]"], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]]]

2001-10-29