Abs[Sin[x + I y]] == Sqrt[Cosh[y]^2 Sin[x]^2 + Cos[x]^2 Sinh[y]^2]

Cell[BoxData[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Sin", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", SqrtBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "y", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", "x", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sinh", "[", "y", "]"]], "2"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#10869; </mo> <msqrt> <mrow> <mrow> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <apply> <sin /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <cosh /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <sin /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <cos /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <sinh /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", RowBox[List["Sin", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", SqrtBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "y", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", "x", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sinh", "[", "y", "]"]], "2"]]]]]]]]]]

2001-10-29