If `x = (sqrt2-1 ) ^(-(1)/(2))` then the value of `(x ^(2) - (1)/(x ^(2))) ` is

Options

A. 2
B. `-2 sqrt2`
C. `2 sqrt2`
D. `-sqrt2`

Answer

To solve the problem, we start with the given expression for \( x \):

\[ x = (\sqrt{2} - 1)^{-\frac{1}{2}} \]

### Step 1: Rewrite \( x \)

We can rewrite \( x \) as:

\[ x = \frac{1}{\sqrt{\sqrt{2} - 1}} \]

### Step 2: Calculate \( x^2 \)

Now, we square \( x \):

\[ x^2 = \left(\frac{1}{\sqrt{\sqrt{2} - 1}}\right)^2 = \frac{1}{\sqrt{2} - 1} \]

### Step 3: Find \( \frac{1}{x^2} \)

Next, we find \( \frac{1}{x^2} \):

\[ \frac{1}{x^2} = \sqrt{2} - 1 \]

### Step 4: Calculate \( x^2 - \frac{1}{x^2} \)

Now, we can find \( x^2 - \frac{1}{x^2} \):

\[ x^2 - \frac{1}{x^2} = \frac{1}{\sqrt{2} - 1} - (\sqrt{2} - 1) \]

### Step 5: Simplify the expression

To simplify this expression, we need a common denominator:

\[ x^2 - \frac{1}{x^2} = \frac{1 - (\sqrt{2} - 1)(\sqrt{2} - 1)}{\sqrt{2} - 1} \]

Calculating \( (\sqrt{2} - 1)(\sqrt{2} - 1) \):

\[ (\sqrt{2} - 1)(\sqrt{2} - 1) = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

Substituting back, we have:

\[ x^2 - \frac{1}{x^2} = \frac{1 - (3 - 2\sqrt{2})}{\sqrt{2} - 1} = \frac{1 - 3 + 2\sqrt{2}}{\sqrt{2} - 1} = \frac{-2 + 2\sqrt{2}}{\sqrt{2} - 1} \]

### Step 6: Factor out the common term

Factoring out the common term in the numerator:

\[ = \frac{2(\sqrt{2} - 1)}{\sqrt{2} - 1} \]

### Step 7: Cancel the common terms

Since \( \sqrt{2} - 1 \) is not zero, we can cancel it out:

\[ = 2 \]

Thus, the final value of \( x^2 - \frac{1}{x^2} \) is:

\[ \boxed{2} \]