ArcCsc[x] - ArcCot[y] == ((x Sqrt[1 + 1/y^2] y Sqrt[(-1 + x^2 - 2 Sqrt[1 - 1/x^2] x y + y^2)/ (x^2 (1 + y^2))])/(Sqrt[1 - 1/x^2] x - y)) ArcSin[(Sqrt[1 - 1/x^2] x + 1/y)/(x Sqrt[1 + 1/y^2])] + (Pi (1 - (Sqrt[1 - 1/x^2] x)/y))/(2 x Sqrt[1 + 1/y^2] Sqrt[(Sqrt[1 - 1/x^2] x - y)^2/(x^2 (1 + y^2))]) + Pi (1 + (1 - (Sqrt[1 - 1/x^2] x)/y)/(x Sqrt[1 + 1/y^2] Sqrt[(Sqrt[1 - 1/x^2] x - y)^2/(x^2 (1 + y^2))])) Floor[(Arg[Sqrt[1 - 1/x^2] + I/x] + Arg[(I + 1/y)/Sqrt[1 + 1/y^2]])/ (2 Pi)] - Pi (1 + (x Sqrt[1 + 1/y^2] y Sqrt[(-1 + x^2 - 2 Sqrt[1 - 1/x^2] x y + y^2)/ (x^2 (1 + y^2))])/(Sqrt[1 - 1/x^2] x - y)) Floor[-((-Pi + Arg[Sqrt[1 - 1/x^2] + I/x] + Arg[(I + 1/y)/Sqrt[1 + 1/y^2]])/(2 Pi))]

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&#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <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> <mtext> </mtext> </mrow> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mi> y </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + 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</msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> x </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <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> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <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> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mi> y </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> &#8520; </mi> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> 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2007-05-02