The value of `((x/y- y/x)(y/z -z/y) (z/x -x/z))/((1/x^(2) -1/y^(2))(1/y^(2)-1/z^(2))(1/z^(2)-1/x^(2)))` is

Options

A. `x^(2)y^(2)z^(2)`
B. `-x^(2)y^(2)z^(2)`
C. `-x^(2)y^(2)`
D. None of these

Answer

To solve the given expression

\[ \frac{( \frac{x}{y} - \frac{y}{x})( \frac{y}{z} - \frac{z}{y})( \frac{z}{x} - \frac{x}{z})}{( \frac{1}{x^2} - \frac{1}{y^2})( \frac{1}{y^2} - \frac{1}{z^2})( \frac{1}{z^2} - \frac{1}{x^2})} \]

we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify the Numerator

The numerator consists of three products:

1. \(\frac{x}{y} - \frac{y}{x} = \frac{x^2 - y^2}{xy}\)
2. \(\frac{y}{z} - \frac{z}{y} = \frac{y^2 - z^2}{yz}\)
3. \(\frac{z}{x} - \frac{x}{z} = \frac{z^2 - x^2}{zx}\)

Thus, the numerator can be rewritten as:

\[ \frac{x^2 - y^2}{xy} \cdot \frac{y^2 - z^2}{yz} \cdot \frac{z^2 - x^2}{zx} \]

Combining these fractions gives:

\[ \frac{(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)}{xyz \cdot xyz} = \frac{(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)}{(xyz)^2} \]

### Step 2: Simplify the Denominator

The denominator consists of three differences of reciprocals:

1. \(\frac{1}{x^2} - \frac{1}{y^2} = \frac{y^2 - x^2}{x^2y^2}\)
2. \(\frac{1}{y^2} - \frac{1}{z^2} = \frac{z^2 - y^2}{y^2z^2}\)
3. \(\frac{1}{z^2} - \frac{1}{x^2} = \frac{x^2 - z^2}{z^2x^2}\)

Thus, the denominator can be rewritten as:

\[ \frac{(y^2 - x^2)(z^2 - y^2)(x^2 - z^2)}{(x^2y^2)(y^2z^2)(z^2x^2)} \]

Combining these fractions gives:

\[ \frac{(y^2 - x^2)(z^2 - y^2)(x^2 - z^2)}{(x^2y^2z^2)^2} \]

### Step 3: Combine the Numerator and Denominator

Now we can combine the simplified numerator and denominator:

\[ \frac{\frac{(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)}{(xyz)^2}}{\frac{(y^2 - x^2)(z^2 - y^2)(x^2 - z^2)}{(x^2y^2z^2)^2}} \]

This simplifies to:

\[ \frac{(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)}{(y^2 - x^2)(z^2 - y^2)(x^2 - z^2)} \cdot \frac{(x^2y^2z^2)^2}{(xyz)^2} \]

### Step 4: Cancel Common Terms

The terms \((x^2 - y^2)\), \((y^2 - z^2)\), and \((z^2 - x^2)\) will cancel out, leading to:

\[ \frac{(x^2y^2z^2)}{1} = x^2y^2z^2 \]

### Step 5: Final Result

The final expression simplifies to:

\[ -1 \cdot \frac{1}{1} = -x^2y^2z^2 \]

Thus, the value of the given expression is:

\[ -x^2y^2z^2 \]