A sub-aquatic drone's descent angle θ \theta\,θ is controlled by a buoyancy system. To maintain stable movement, the angle must satisfy the equilibrium equation:
5cot2θ+3 cosec2θ=2 cosec θ+10 5\cot^2\theta + 3\text{ cosec}^2\theta = 2\text{ cosec }\theta + 10 5cot2θ+3 cosec2θ=2 cosec θ+10Show that this equation can be written in the form
a cosec2θ+b cosec θ+c=0 a\text{ cosec}^2\theta + b\text{ cosec }\theta + c = 0 a cosec2θ+b cosec θ+c=0where aaa, bbb, and c c\,c are integers to be found.
Hence, given that the drone is tilted at an obtuse angle θ \theta\,θ that satisfies this stability equation, determine the exact value of tanθ\tan \thetatanθ. Fully justify your answer.
