If `|2 sin theta-cosec theta| ge 1` and `theta ne (n pi)/2, n in Z`, then

Options

A. `cos 2 theta ge 1//2`
B. `cos 2 theta ge 1//4`
C. `cos 2 theta le 1//2`
D. `cos 2 theta le 1//4`

Explanation

To solve the inequality \( |2 \sin \theta - \csc \theta| \geq 1 \) where \( \theta \neq \frac{n \pi}{2}, n \in \mathbb{Z} \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ |2 \sin \theta - \csc \theta| \geq 1 \]
Recall that \( \csc \theta = \frac{1}{\sin \theta} \). Thus, we can rewrite the expression:
\[ |2 \sin \theta - \frac{1}{\sin \theta}| \geq 1 \]

### Step 2: Eliminate the absolute value
This gives us two cases to consider:
1. \( 2 \sin \theta - \frac{1}{\sin \theta} \geq 1 \)
2. \( 2 \sin \theta - \frac{1}{\sin \theta} \leq -1 \)

### Step 3: Solve Case 1
For the first case:
\[ 2 \sin \theta - \frac{1}{\sin \theta} \geq 1 \]
Multiply through by \( \sin \theta \) (noting \( \sin \theta > 0 \)):
\[ 2 \sin^2 \theta - 1 \geq \sin \theta \]
Rearranging gives:
\[ 2 \sin^2 \theta - \sin \theta - 1 \geq 0 \]

### Step 4: Factor the quadratic
We can factor the quadratic:
\[ (2 \sin \theta + 1)(\sin \theta - 1) \geq 0 \]
The critical points are \( \sin \theta = -\frac{1}{2} \) and \( \sin \theta = 1 \).

### Step 5: Analyze the intervals
We analyze the sign of the product in the intervals determined by the critical points:
1. \( \sin \theta < -\frac{1}{2} \): The product is negative.
2. \( -\frac{1}{2} < \sin \theta < 1 \): The product is positive.
3. \( \sin \theta = 1 \): The product is zero.

Thus, the solution for this case is:
\[ -\frac{1}{2} < \sin \theta \leq 1 \]

### Step 6: Solve Case 2
For the second case:
\[ 2 \sin \theta - \frac{1}{\sin \theta} \leq -1 \]
Again, multiply through by \( \sin \theta \):
\[ 2 \sin^2 \theta + 1 \leq \sin \theta \]
Rearranging gives:
\[ 2 \sin^2 \theta - \sin \theta + 1 \leq 0 \]
This quadratic has no real roots (discriminant \( (-1)^2 - 4 \cdot 2 \cdot 1 < 0 \)), so it is always positive. Thus, there are no solutions from this case.

### Step 7: Combine results
The only valid solutions come from Case 1:
\[ -\frac{1}{2} < \sin \theta \leq 1 \]

### Conclusion
Thus, the solution set for the inequality \( |2 \sin \theta - \csc \theta| \geq 1 \) is:
\[ \sin \theta \in \left(-\frac{1}{2}, 1\right] \]