ArcCsc[x] + ArcCsc[y] == Sign[x] Pi - ArcSin[Sqrt[1 - 1/x^2]/y + Sqrt[1 - 1/y^2]/x] /; x y > 0 && 1/x^2 + 1/y^2 > 1

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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> <mrow> <mrow> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arccsc /> <ci> x </ci> </apply> <apply> <arccsc /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Sign </ci> <ci> x </ci> </apply> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arcsin /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCsc", "[", "x_", "]"]], "+", RowBox[List["ArcCsc", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Sign", "[", "x", "]"]], " ", "\[Pi]"]], "-", RowBox[List["ArcSin", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "y"], "+", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "x"]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["x", " ", "y"]], ">", "0"]], "&&", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["x", "2"]], "+", FractionBox["1", SuperscriptBox["y", "2"]]]], ">", "1"]]]]]]]]]]

2001-10-29