If
`abs(cos x)^(sin^2 x - 3/2 sin x + 1/2) = 1`,
then possible values of x are

Options

A. `n pi or n pi + (-1)^n pi //6, n in I`
B. `n pi or 2 n pi + pi//2or npi + (-1)^n pi//6, n in I`
C. `n pi + (-1)^n pi //6, n in I`.
D. `n pi , n in I`

Answer

To solve the equation \( |\cos x|^{\sin^2 x - \frac{3}{2} \sin x + \frac{1}{2}} = 1 \), we can analyze the conditions under which the expression equals 1.

### Step 1: Analyze the equation
The expression \( |a|^b = 1 \) can be true under the following conditions:
1. \( |a| = 1 \) (i.e., \( a = 1 \) or \( a = -1 \))
2. \( b = 0 \) (i.e., the exponent is zero)

### Step 2: Case 1 - \( |\cos x| = 1 \)
If \( |\cos x| = 1 \), then \( \cos x = 1 \) or \( \cos x = -1 \).

- **For \( \cos x = 1 \)**:
\[ x = 2n\pi \quad (n \in \mathbb{Z}) \]

- **For \( \cos x = -1 \)**:
\[ x = (2n + 1)\pi \quad (n \in \mathbb{Z}) \]

### Step 3: Case 2 - \( \sin^2 x - \frac{3}{2} \sin x + \frac{1}{2} = 0 \)
Next, we solve the quadratic equation:
\[ \sin^2 x - \frac{3}{2} \sin x + \frac{1}{2} = 0 \]
Using the quadratic formula:
\[ \sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -\frac{3}{2}, c = \frac{1}{2} \).

Calculating the discriminant:
\[ b^2 - 4ac = \left(-\frac{3}{2}\right)^2 - 4 \cdot 1 \cdot \frac{1}{2} = \frac{9}{4} - 2 = \frac{1}{4} \]

Now substituting into the quadratic formula:
\[ \sin x = \frac{\frac{3}{2} \pm \frac{1}{2}}{2} \]
This gives us:
1. \( \sin x = \frac{2}{2} = 1 \)
2. \( \sin x = \frac{1}{2} \)

### Step 4: Solve for \( \sin x = 1 \)
For \( \sin x = 1 \):
\[ x = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \]
However, we cannot use this solution because at \( x = \frac{\pi}{2} \), \( \cos x = 0 \), which does not satisfy \( |\cos x| = 1 \).

### Step 5: Solve for \( \sin x = \frac{1}{2} \)
For \( \sin x = \frac{1}{2} \):
\[ x = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad x = \frac{5\pi}{6} + 2n\pi \quad (n \in \mathbb{Z}) \]

### Final Solution
Combining all the results, the possible values of \( x \) are:
1. \( x = 2n\pi \) (from \( \cos x = 1 \))
2. \( x = (2n + 1)\pi \) (from \( \cos x = -1 \))
3. \( x = \frac{\pi}{6} + 2n\pi \)
4. \( x = \frac{5\pi}{6} + 2n\pi \)