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http://functions.wolfram.com/01.12.16.0030.01
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ArcSin[x] + ArcSin[y] == Sign[x + y] ArcCos[Sqrt[1 - x^2] Sqrt[1 - y^2] -
x y] /; (x > -1 && y > -1) || (x < 1 && y < 1)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", "x", "]"]], "+", RowBox[List["ArcSin", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["Sign", "[", RowBox[List["x", "+", "y"]], "]"]], " ", RowBox[List["ArcCos", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "-", RowBox[List["x", " ", "y"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["x", ">", RowBox[List["-", "1"]]]], "&&", RowBox[List["y", ">", RowBox[List["-", "1"]]]]]], "||", RowBox[List[RowBox[List["x", "<", "1"]], "&&", RowBox[List["y", "<", "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> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> - </mo> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> x </mi> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> y </mi> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> x </mi> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> y </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arcsin /> <ci> x </ci> </apply> <apply> <arcsin /> <ci> y </ci> </apply> </apply> <apply> <times /> <apply> <ci> Sign </ci> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> </apply> <apply> <arccos /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <gt /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <apply> <gt /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <lt /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcSin", "[", "x_", "]"]], "+", RowBox[List["ArcSin", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Sign", "[", RowBox[List["x", "+", "y"]], "]"]], " ", RowBox[List["ArcCos", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "-", RowBox[List["x", " ", "y"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["x", ">", RowBox[List["-", "1"]]]], "&&", RowBox[List["y", ">", RowBox[List["-", "1"]]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["x", "<", "1"]], "&&", RowBox[List["y", "<", "1"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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