If cosec^-1x + cosec^-1y + cosec^-1z = -(3π)/2, then x/y+y/z+z/x=

We know that the minimum value of $cosec^{-1} x$ is $-\frac{\pi}{2}$ which is attained at x = -1

∴ $cosec^{-1} x + cosec^{-1}y + cosec^{-1} z=-\frac{3\pi}{2}$

$⇒ cosec^{-1} x + cosec^{-1}+cosec^{-1}z = \left(-\frac{\pi}{2}\right)+\left(-\frac{\pi}{2}\right)+\left(-\frac{\pi}{2}\right)$

$⇒ cosec^{-1}x=-\frac{\pi}{2}, cosec^{-1}y = -\frac{\pi}{2}, cosec^{-1}z =-\frac{\pi}{2}$

$ ⇒x = -1,  y = -1, z = -1 $

$∴\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{(-1)}{(-1)}+\frac{(-1)}{(-1)}+\frac{(-1)}{(-1)}= 3$