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

Options

A. 1
B. `-3`
C. 3
D. `3/2`

Explanation

To solve the equation \( \csc^{-1}x + \csc^{-1}y + \csc^{-1}z = -\frac{3\pi}{2} \), we will follow these steps:

### Step 1: Rewrite the equation using sine
We know that \( \csc^{-1}x = \sin^{-1}\left(\frac{1}{x}\right) \). Therefore, we can rewrite the equation as:
\[ \sin^{-1}\left(\frac{1}{x}\right) + \sin^{-1}\left(\frac{1}{y}\right) + \sin^{-1}\left(\frac{1}{z}\right) = -\frac{3\pi}{2} \]

### Step 2: Use the property of sine inverse
The sine function is periodic, and \( \sin^{-1}(a) + \sin^{-1}(b) + \sin^{-1}(c) \) can be transformed using the identity:
\[ \sin^{-1}(a) + \sin^{-1}(b) = \sin^{-1}\left(a\sqrt{1-b^2} + b\sqrt{1-a^2}\right) \]
However, since we have a negative angle, we can express it as:
\[ \sin^{-1}\left(\frac{1}{x}\right) + \sin^{-1}\left(\frac{1}{y}\right) + \sin^{-1}\left(\frac{1}{z}\right) = \sin^{-1}\left(-1\right) \]
This implies that:
\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = -1 \]

### Step 3: Use the identity for the sum of reciprocals
From the above step, we can write:
\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = -1 \]
This can be rearranged to:
\[ \frac{yz + zx + xy}{xyz} = -1 \]
Thus, we have:
\[ yz + zx + xy = -xyz \]

### Step 4: Find the required expression
We need to find \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \). This can be rewritten as:
\[ \frac{x^2z + y^2x + z^2y}{xyz} \]
Using the earlier result \( yz + zx + xy = -xyz \), we can substitute:
\[ \frac{x^2z + y^2x + z^2y}{xyz} = -\frac{xyz}{xyz} = -1 \]

### Final Result
Thus, the value of \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \) is:
\[ \boxed{-1} \]