`[(x + 1)/(x - 1) - (x - 1)/(x + 1) - (4x)/(x^2 + 1)] div 4/(x^4 - 1)` when simplified is equal to :

Options

A. `1 `
B. `0 `
C. `x^2-1`
D. `2`

Answer

To simplify the expression \(\left[\frac{x + 1}{x - 1} - \frac{x - 1}{x + 1} - \frac{4x}{x^2 + 1}\right] \div \frac{4}{x^4 - 1}\), we will follow these steps:

### Step 1: Simplify the expression inside the brackets
We start with the expression:
\[ \frac{x + 1}{x - 1} - \frac{x - 1}{x + 1} \]
To combine these fractions, we need a common denominator, which is \((x - 1)(x + 1)\):
\[ \frac{(x + 1)^2 - (x - 1)^2}{(x - 1)(x + 1)} \]

### Step 2: Expand the numerators
Now, we expand \((x + 1)^2\) and \((x - 1)^2\):
\[ (x + 1)^2 = x^2 + 2x + 1 \]
\[ (x - 1)^2 = x^2 - 2x + 1 \]
Subtracting these gives:
\[ (x^2 + 2x + 1) - (x^2 - 2x + 1) = 4x \]
Thus, we have:
\[ \frac{4x}{(x - 1)(x + 1)} \]

### Step 3: Combine with the third term
Now we need to subtract \(\frac{4x}{x^2 + 1}\) from \(\frac{4x}{(x - 1)(x + 1)}\):
\[ \frac{4x}{(x - 1)(x + 1)} - \frac{4x}{x^2 + 1} \]
The common denominator here is \((x - 1)(x + 1)(x^2 + 1)\). Thus, we rewrite both fractions:
\[ \frac{4x(x^2 + 1) - 4x(x - 1)(x + 1)}{(x - 1)(x + 1)(x^2 + 1)} \]

### Step 4: Simplify the numerator
Now we simplify the numerator:
\[ 4x(x^2 + 1) - 4x((x^2 - 1)) = 4x(x^2 + 1 - x^2 + 1) = 4x(2) = 8x \]
So, we have:
\[ \frac{8x}{(x - 1)(x + 1)(x^2 + 1)} \]

### Step 5: Divide by \(\frac{4}{x^4 - 1}\)
Now we need to divide this by \(\frac{4}{x^4 - 1}\):
\[ \frac{8x}{(x - 1)(x + 1)(x^2 + 1)} \div \frac{4}{x^4 - 1} \]
This is equivalent to multiplying by the reciprocal:
\[ \frac{8x}{(x - 1)(x + 1)(x^2 + 1)} \cdot \frac{x^4 - 1}{4} \]

### Step 6: Simplify the expression
Now we can simplify:
\[ \frac{8x(x^4 - 1)}{4(x - 1)(x + 1)(x^2 + 1)} = \frac{2x(x^4 - 1)}{(x - 1)(x + 1)(x^2 + 1)} \]

### Step 7: Factor \(x^4 - 1\)
We can factor \(x^4 - 1\) as \((x^2 - 1)(x^2 + 1)\), and further factor \(x^2 - 1\) as \((x - 1)(x + 1)\):
\[ x^4 - 1 = (x - 1)(x + 1)(x^2 + 1) \]
Substituting this back gives:
\[ \frac{2x \cdot (x - 1)(x + 1)(x^2 + 1)}{(x - 1)(x + 1)(x^2 + 1)} \]
The \((x - 1)(x + 1)(x^2 + 1)\) terms cancel out:
\[ 2x \]

### Final Result
Thus, the simplified expression is:
\[ \boxed{2x} \]