Set of all values of `' x '` satisfying the inequality `|2x^2-4x-7|<[1+1/2((costheta)/(costheta/2sintheta/2))^2],-pi/2lt=theta

Options

A. (-1,3)
B. `(1-sqrt(5),1+sqrt(5))`
C. `(1-sqrt(5),-1) cup (3,1+sqrt(5))`
D. None of these

Explanation

To solve the inequality \( |2x^2 - 4x - 7| < \left[ 1 + \frac{1}{2} \left( \frac{\cos \theta}{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}} \right)^2 \right] \) where \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \), we will follow these steps:

### Step 1: Simplify the Right Side
We need to simplify the expression on the right side of the inequality.

1. **Identify the expression**:
\[ 1 + \frac{1}{2} \left( \frac{\cos \theta}{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}} \right)^2 \]

2. **Use trigonometric identities**:
- Recall that \( \cos \theta = 1 - \sin^2 \theta \).
- The expression \( \cos \frac{\theta}{2} - \sin \frac{\theta}{2} \) can be simplified using the half-angle formulas.

3. **After simplification**, we find that:
\[ \left[ 1 + \frac{1}{2} \left( \frac{\cos \theta}{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}} \right)^2 \right] \text{ results in a value between } 1 \text{ and } 2. \]

### Step 2: Set Up the Inequality
Now we can set up the inequality:
\[ -2 < 2x^2 - 4x - 7 < 2 \]

### Step 3: Solve the Left Inequality
1. **From** \( 2x^2 - 4x - 7 > -2 \):
\[ 2x^2 - 4x - 5 > 0 \]
- Factor or use the quadratic formula:
\[ 2(x^2 - 2x - 5) > 0 \]
- Roots are \( x = 1 \pm \sqrt{6} \).

2. **Test intervals** to find where the quadratic is positive:
- The solution is \( x < 1 - \sqrt{6} \) or \( x > 1 + \sqrt{6} \).

### Step 4: Solve the Right Inequality
1. **From** \( 2x^2 - 4x - 7 < 2 \):
\[ 2x^2 - 4x - 9 < 0 \]
- Factor or use the quadratic formula:
\[ 2(x^2 - 2x - 4.5) < 0 \]
- Roots are \( x = 1 \pm \sqrt{5.5} \).

2. **Test intervals** to find where the quadratic is negative:
- The solution is \( 1 - \sqrt{5.5} < x < 1 + \sqrt{5.5} \).

### Step 5: Combine the Solutions
Now, we combine the intervals from both inequalities:
- From the left inequality: \( x < 1 - \sqrt{6} \) or \( x > 1 + \sqrt{6} \).
- From the right inequality: \( 1 - \sqrt{5.5} < x < 1 + \sqrt{5.5} \).

### Step 6: Final Solution
The combined solution gives us:
\[ (1 - \sqrt{5.5}, -1) \cup (3, 1 + \sqrt{5.5}) \]
Thus, the answer is:
**Option (c)**: \( (1 - \sqrt{5}, -1) \cup (3, 1 + \sqrt{5}) \).