ArcSin[x] + ArcCos[y] == Pi/2 + Sign[x - y] ArcCos[x y + Sqrt[1 - x^2] Sqrt[1 - y^2]] /; -1 < x < 1 || -1 < y < 1 || x y < 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", "x", "]"]], "+", RowBox[List["ArcCos", "[", "y", "]"]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["x", "-", "y"]], "]"]], " ", RowBox[List["ArcCos", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "1"]], "||", RowBox[List[RowBox[List["-", "1"]], "<", "y", "<", "1"]], "||", RowBox[List[RowBox[List["x", " ", "y"]], "<", "0"]]]]]]]]

<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> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> y </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arccos /> <ci> y </ci> </apply> <apply> <arcsin /> <ci> x </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <arccos /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <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> </apply> <apply> <ci> Sign </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

2001-10-29