`-1lt cos(2x -(pi)/(3))lt(1)/(2)` has solution set (A) `x in ((pi)/(3)+2n pi, (5pi)/(3)+2n pi), n in I` (B) `x in ((pi)/(3)+2n pi, (2pi)/(3)+2n pi)uu((2pi)/(3)+2n pi, (2n+1)pi), n in I` (C) `x in ((pi)/(3)+2n pi, pi+2n pi)uu(pi+2n pi, (5pi)/(3)+2n pi), n in I` (D) `x in ((pi)/(3)+n pi, (2pi)/(3)+n pi)uu((2pi)/(3)+n pi, (n+1)pi), n in I`

Options

A. `x in ((pi)/(3)+2n pi, (5pi)/(3)+2n pi), n in I`
B. `x in ((pi)/(3)+2n pi, (2pi)/(3)+2n pi)uu((2pi)/(3)+2n pi, (2n+1)pi), n in I`
C. `x in ((pi)/(3)+2n pi, pi+2n pi)uu(pi+2n pi, (5pi)/(3)+2n pi), n in I`
D. `x in ((pi)/(3)+n pi, (2pi)/(3)+n pi)uu((2pi)/(3)+n pi, (n+1)pi), n in I`

Explanation

To solve the inequality \(-1 < \cos(2x - \frac{\pi}{3}) < \frac{1}{2}\), we will break it down step by step.

### Step 1: Understanding the Range of Cosine
The cosine function ranges from -1 to 1. Therefore, we need to find the angles for which \(\cos(\theta) = -1\) and \(\cos(\theta) = \frac{1}{2}\).

### Step 2: Finding Angles for the Inequalities
1. \(\cos(2x - \frac{\pi}{3}) = -1\) occurs at:
\[ 2x - \frac{\pi}{3} = \pi + 2n\pi \quad \text{(for integer } n\text{)} \]
Rearranging gives:
\[ 2x = \pi + \frac{\pi}{3} + 2n\pi = \frac{4\pi}{3} + 2n\pi \]
\[ x = \frac{2\pi}{3} + n\pi \]

2. \(\cos(2x - \frac{\pi}{3}) = \frac{1}{2}\) occurs at:
\[ 2x - \frac{\pi}{3} = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad 2x - \frac{\pi}{3} = \frac{5\pi}{3} + 2n\pi \]
For the first case:
\[ 2x = \frac{\pi}{3} + \frac{\pi}{3} + 2n\pi = \frac{2\pi}{3} + 2n\pi \]
\[ x = \frac{\pi}{3} + n\pi \]
For the second case:
\[ 2x = \frac{5\pi}{3} + \frac{\pi}{3} + 2n\pi = 2\pi + 2n\pi \]
\[ x = \pi + n\pi \]

### Step 3: Combining the Results
From the inequalities, we have:
1. \(x\) must be greater than \(\frac{\pi}{3} + n\pi\)
2. \(x\) must be less than \(\frac{2\pi}{3} + n\pi\) and less than \(\pi + n\pi\)

### Step 4: Writing the Final Solution Set
The solution set can be expressed as:
\[ x \in \left(\frac{\pi}{3} + n\pi, \frac{2\pi}{3} + n\pi\right) \cup \left(\frac{2\pi}{3} + n\pi, \pi + n\pi\right) \cup \left(\pi + n\pi, \frac{5\pi}{3} + n\pi\right) \]

### Step 5: Conclusion
The correct option that matches our derived solution is:
\[ \text{(C) } x \in \left(\frac{\pi}{3} + n\pi, \pi + n\pi\right) \cup \left(\pi + n\pi, \frac{5\pi}{3} + n\pi\right), \, n \in \mathbb{Z} \]