ArcTan[x] - ArcCsc[y] == ArcTan[(Sqrt[1 + x^2] Sqrt[1 - (Sqrt[1 - 1/y^2] + x/y)^2/(1 + x^2)])/ ((-1)^Floor[1/2 - Arg[(x Sqrt[1 - 1/y^2] - 1/y)/Sqrt[1 + x^2]]/Pi] (Sqrt[1 - 1/y^2] + x/y))] + (1/2) Pi ((-1)^(Floor[1/2 + Arg[1 + x^2]/(2 Pi) - Arg[x Sqrt[1 - 1/y^2] - 1/y]/Pi] - 2 Floor[1/2 + Arg[1 + x^2]/(4 Pi) - Arg[x Sqrt[1 - 1/y^2] - 1/y]/ (2 Pi)] + Floor[1/2 - Arg[1 + x^2]/(2 Pi) + Arg[-Sqrt[1 - 1/y^2] - x/y]/Pi]) + (-1)^Floor[1/2 - Arg[(x Sqrt[1 - 1/y^2] - 1/y)/Sqrt[1 + x^2]]/Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[(x Sqrt[1 - 1/y^2] - 1/y)/Sqrt[1 + x^2]]/Pi]) Floor[(Arg[(I - x)/Sqrt[1 + x^2]] + Arg[Sqrt[1 - 1/y^2] - I/y])/ (2 Pi)] - 2 (-1 + (-1)^Floor[1/2 - Arg[(x Sqrt[1 - 1/y^2] - 1/y)/Sqrt[1 + x^2]]/ Pi]) Floor[1/2 - (Arg[(I - x)/Sqrt[1 + x^2]] + Arg[Sqrt[1 - 1/y^2] - I/y])/(2 Pi)])

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