If `x ne pm 1`, then `(x+1)/(x-1)-(x-1)/(x+1)=`

Options

A. `(2x)/(x-1)`
B. `(2x)/(x^(2)+1)`
C. `(2x)/(x^(2)-1)`
D. `(4x)/(x^(2)-1)`

Explanation

To solve the expression \((x+1)/(x-1) - (x-1)/(x+1)\), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the expression**:
\[ \frac{x+1}{x-1} - \frac{x-1}{x+1} \]

2. **Find a common denominator**:
The common denominator for the two fractions is \((x-1)(x+1)\).

3. **Rewrite each fraction with the common denominator**:
\[ \frac{(x+1)(x+1)}{(x-1)(x+1)} - \frac{(x-1)(x-1)}{(x-1)(x+1)} \]
This simplifies to:
\[ \frac{(x+1)^2 - (x-1)^2}{(x-1)(x+1)} \]

4. **Expand the numerators**:
- Expanding \((x+1)^2\):
\[ (x+1)^2 = x^2 + 2x + 1 \]
- Expanding \((x-1)^2\):
\[ (x-1)^2 = x^2 - 2x + 1 \]

5. **Substituting back into the expression**:
\[ \frac{x^2 + 2x + 1 - (x^2 - 2x + 1)}{(x-1)(x+1)} \]

6. **Combine like terms in the numerator**:
\[ x^2 + 2x + 1 - x^2 + 2x - 1 = 4x \]
So, the expression simplifies to:
\[ \frac{4x}{(x-1)(x+1)} \]

7. **Final result**:
The final simplified expression is:
\[ \frac{4x}{x^2 - 1} \]
(since \(x^2 - 1 = (x-1)(x+1)\))

### Final Answer:
\[ \frac{4x}{x^2 - 1} \]