Coth[z/2] == Coth[z] + Csch[z]

Cell[BoxData[RowBox[List[RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], "\[Equal]", RowBox[List[RowBox[List["Coth", "[", "z", "]"]], "+", RowBox[List["Csch", "[", "z", "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <coth /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <coth /> <ci> z </ci> </apply> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Coth", "[", FractionBox["z_", "2"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Coth", "[", "z", "]"]], "+", RowBox[List["Csch", "[", "z", "]"]]]]]]]]

2001-10-29