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   http://functions.wolfram.com/01.13.27.2241.01
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    ArcCos[(2 Sqrt[-1 - z^2])/z^2] == 
 Pi/2 - ((z^3 Sqrt[-2 - z^2] Sqrt[-1 - z^2])/(2 Sqrt[1 - I z] (1 + I z) 
     Sqrt[-2 - 3 z^2 - z^4])) Sqrt[(1 + z^2)/z^4] Sqrt[(I - z)/z] 
   Sqrt[-(I/z)] (Pi (Sqrt[-(1/z^2)] z - I Sqrt[(z + I Sqrt[2])/z] Sqrt[I/z] 
       Sqrt[(-I) z] Sqrt[z/(I Sqrt[2] + z)] + I Sqrt[(z - I Sqrt[2])/z] 
       Sqrt[-(I/z)] Sqrt[I z] Sqrt[z/((-I) Sqrt[2] + z)] - 
      (z^3/(1 + z^2)) Sqrt[-((1 + z^2)/z^2)] Sqrt[(1 + z^2)/z^4]) + 
    4 ArcCsch[z]) 
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   Cell[BoxData[RowBox[List[RowBox[List["ArcCos", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "3"], SqrtBox[RowBox[List[RowBox[List["-", "2"]], "-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]], "  ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["3", " ", SuperscriptBox["z", "2"]]], "-", SuperscriptBox["z", "4"]]]]]]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], "z"]], " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", SqrtBox[FractionBox[RowBox[List["z", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]]]], "z"]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["\[ImaginaryI]", SqrtBox["2"]]], "+", "z"]]]]]], "+", RowBox[List["\[ImaginaryI]", SqrtBox[FractionBox[RowBox[List["z", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]]]], "z"]], " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], SqrtBox["2"]]], "+", "z"]]]]]], "-", RowBox[List[FractionBox[SuperscriptBox["z", "3"], RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]]]]]], " ", ")"]]]], "+", RowBox[List["4", " ", RowBox[List["ArcCsch", "[", "z", "]"]]]]]], ")"]]]]]]]]]] 
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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>  <msup>  <mi> cos </mi>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ( </mo>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mrow>  <mo> - </mo>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mi> π </mi>  <mn> 2 </mn>  </mfrac>  <mo> - </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mi> z </mi>  <mn> 3 </mn>  </msup>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mrow>  <mo> - </mo>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mrow>  <mo> - </mo>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mrow>  <mo> - </mo>  <msup>  <mi> z </mi>  <mn> 4 </mn>  </msup>  </mrow>  <mo> - </mo>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  </msqrt>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <msup>  <mi> z </mi>  <mn> 4 </mn>  </msup>  </mfrac>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mrow>  <mi> ⅈ </mi>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mi> z </mi>  </mfrac>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mo> - </mo>  <mfrac>  <mi> ⅈ </mi>  <mi> z </mi>  </mfrac>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mtext>   </mtext>  <msup>  <mi> z </mi>  <mn> 3 </mn>  </msup>  </mrow>  <mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mfrac>  </mrow>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mfrac>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <msup>  <mi> z </mi>  <mn> 4 </mn>  </msup>  </mfrac>  </msqrt>  </mrow>  <mo> + </mo>  <mrow>  <msqrt>  <mrow>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mfrac>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> + </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mrow>  <mi> z </mi>  <mo> - </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  </mrow>  </mrow>  <mi> z </mi>  </mfrac>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mo> - </mo>  <mfrac>  <mi> ⅈ </mi>  <mi> z </mi>  </mfrac>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mi> z </mi>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mi> ⅈ </mi>  </mrow>  <mo> ⁢ </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  </mrow>  <mo> + </mo>  <mi> z </mi>  </mrow>  </mfrac>  </msqrt>  </mrow>  <mo> - </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  </mrow>  </mrow>  <mi> z </mi>  </mfrac>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mi> ⅈ </mi>  <mi> z </mi>  </mfrac>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mrow>  <mo> - </mo>  <mi> ⅈ </mi>  </mrow>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mfrac>  <mi> z </mi>  <mrow>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  </mrow>  <mo> + </mo>  <mi> z </mi>  </mrow>  </mfrac>  </msqrt>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <mrow>  <msup>  <mi> csch </mi>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <arccos />  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <pi />  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 3 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  </apply>  <cn type='integer'> -2 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <ci> z </ci>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <plus />  <apply>  <times />  <imaginaryi />  <ci> z </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 4 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 3 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  </apply>  </apply>  <cn type='integer'> -2 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 4 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <plus />  <imaginaryi />  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <plus />  <apply>  <times />  <pi />  <apply>  <plus />  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 3 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 4 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <power />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <ci> z </ci>  </apply>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <apply>  <times />  <apply>  <plus />  <ci> z </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <imaginaryi />  <ci> z </ci>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <ci> z </ci>  <apply>  <power />  <apply>  <plus />  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <imaginaryi />  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <ci> z </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <apply>  <times />  <apply>  <plus />  <ci> z </ci>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <imaginaryi />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <imaginaryi />  </apply>  <ci> z </ci>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <ci> z </ci>  <apply>  <power />  <apply>  <plus />  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <ci> z </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 4 </cn>  <apply>  <arccsch />  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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