The angle of rotation ψ\psiψ of a high-precision robotic arm, where 0<ψ<π0 < \psi < \pi0<ψ<π, is determined by the following equilibrium equation:
4cot2ψ+2=4cscψ+13 4\cot^2 \psi + 2 = 4\csc \psi + 13 4cot2ψ+2=4cscψ+13Show that the equation can be written in the form
acsc2ψ+bcscψ+c=0 a\csc^2 \psi + b\csc \psi + c = 0 acsc2ψ+bcscψ+c=0where aaa, bbb, and ccc are integers to be found.
Hence, given that the robotic arm is positioned at an obtuse angle ψ\psiψ that satisfies the original equation, find the exact value of tanψ\tan \psitanψ. Fully justify your answer.
