`x^(2)+1/(x^(2))=` ____

Options

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

Answer

To solve the equation \( x^2 + \frac{1}{x^2} \), we can use the identity related to the square of a binomial. Here’s the step-by-step solution:

### Step 1: Use the identity
We know that:
\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
From this, we can express \( x^2 + \frac{1}{x^2} \) as:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]

### Step 2: Substitute \( a = x + \frac{1}{x} \)
Let \( a = x + \frac{1}{x} \). Then we can rewrite the equation as:
\[ x^2 + \frac{1}{x^2} = a^2 - 2 \]

### Step 3: Final expression
Thus, we have:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]

### Conclusion
So, the expression \( x^2 + \frac{1}{x^2} \) can be represented as:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]