GoldenRatio == (1/2) (1 + Sqrt[5]) == Exp[ArcCsch[2]]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <exp /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List["GoldenRatio", "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]]

2001-10-29