Step 1: Understanding the Concept:
This question connects the algebraic expression $x + \frac{1}{x}$ with the trigonometric function $\cos\theta$. The key is to know the range of values that both of these expressions can take.
Step 2: Key Formula or Approach:
We need to use two important range properties:
1. For any non-zero real number $x$, the value of $x + \frac{1}{x}$ is always in the interval $(-\infty, -2] \cup [2, \infty)$. This means $|x + \frac{1}{x}| \ge 2$.
2. The range of the cosine function is $[-1, 1]$. This means $|\cos\theta| \le 1$.
Step 3: Detailed Explanation:
We are given the equation:
\[ x + \frac{1}{x} = 2 \cos\theta \]
From the property of the expression on the left side, we know that:
\[ |x + \frac{1}{x}| \ge 2 \]
Substituting the given equation into this inequality:
\[ |2 \cos\theta| \ge 2 \]
\[ 2 |\cos\theta| \ge 2 \]
\[ |\cos\theta| \ge 1 \]
Now, from the property of the cosine function, we know that:
\[ |\cos\theta| \le 1 \]
We have two conditions for $|\cos\theta|$: it must be greater than or equal to 1, and it must be less than or equal to 1. The only value that satisfies both conditions simultaneously is:
\[ |\cos\theta| = 1 \]
This implies that:
\[ \cos\theta = 1 \quad \text{or} \quad \cos\theta = -1 \]
So, $\cos\theta = \pm 1$.
Step 4: Final Answer:
The only possible values for $\cos\theta$ are $\pm 1$. Therefore, option (A) is the correct answer.