If `(x+(1)/(x))=2`, then the value of `(x^(99)+(1)/(x^(99))-2)` is :

Options

A. `-2`
B. 0
C. 2
D. 4

Answer

To solve the problem, we start with the given equation:

1. **Given:**
\[ x + \frac{1}{x} = 2 \]

2. **Rearranging the equation:**
We can square both sides to find a relationship involving \(x^2\):
\[ \left(x + \frac{1}{x}\right)^2 = 2^2 \]
Expanding the left side:
\[ x^2 + 2 + \frac{1}{x^2} = 4 \]
Simplifying this gives:
\[ x^2 + \frac{1}{x^2} = 4 - 2 = 2 \]

3. **Finding \(x^2\):**
Now, we can express \(x^2\) in terms of \(x\):
\[ x^2 + \frac{1}{x^2} = 2 \]

4. **Finding \(x\):**
We can rewrite the equation:
\[ x^2 - 2x + 1 = 0 \]
This can be factored as:
\[ (x - 1)^2 = 0 \]
Thus, we find:
\[ x = 1 \]

5. **Calculating \(x^{99} + \frac{1}{x^{99}} - 2\):**
Now, substituting \(x = 1\) into the expression we need to evaluate:
\[ x^{99} + \frac{1}{x^{99}} - 2 = 1^{99} + \frac{1}{1^{99}} - 2 \]
Since \(1^{99} = 1\) and \(\frac{1}{1^{99}} = 1\), we have:
\[ 1 + 1 - 2 = 0 \]

6. **Final answer:**
Therefore, the value of \(x^{99} + \frac{1}{x^{99}} - 2\) is:
\[ \boxed{0} \]