The expression `cos 3 theta + sin 3 theta + (2 sin 2 theta-3) (sin theta- cos theta)` is positive for all `theta` in

Options

A. `(2 n pi-(3pi)/4, 2npi+pi/4), n in Z`
B. `(2npi-pi/4, 2npi +pi/6), n in Z`
C. `(2npi-pi/3, 2npi +pi/3), n in Z`
D. `(2npi-pi/4, 2npi+(3pi)/4), n in Z`

Explanation

To solve the expression \( \cos 3\theta + \sin 3\theta + (2\sin 2\theta - 3)(\sin \theta - \cos \theta) \) and determine for which values of \( \theta \) it is positive, we will follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
We know the formulas for \( \cos 3\theta \), \( \sin 3\theta \), and \( \sin 2\theta \):
- \( \cos 3\theta = 4\cos^3\theta - 3\cos\theta \)
- \( \sin 3\theta = 3\sin\theta - 4\sin^3\theta \)
- \( \sin 2\theta = 2\sin\theta\cos\theta \)

Substituting these into the expression gives:
\[ \cos 3\theta + \sin 3\theta = (4\cos^3\theta - 3\cos\theta) + (3\sin\theta - 4\sin^3\theta) \]
Thus, the expression becomes:
\[ (4\cos^3\theta - 3\cos\theta + 3\sin\theta - 4\sin^3\theta) + (2(2\sin\theta\cos\theta) - 3)(\sin\theta - \cos\theta) \]

### Step 2: Simplify the expression
Combining the terms, we can rewrite the expression as:
\[ 4\cos^3\theta - 4\sin^3\theta + 3\sin\theta - 3\cos\theta + 4\sin\theta\cos\theta - 3(\sin\theta - \cos\theta) \]
This simplifies to:
\[ 4(\cos^3\theta - \sin^3\theta) + 4\sin\theta\cos\theta - 6\sin\theta + 3\cos\theta \]

### Step 3: Factor the expression
Notice that \( \cos^3\theta - \sin^3\theta \) can be factored using the difference of cubes:
\[ \cos^3\theta - \sin^3\theta = (\cos\theta - \sin\theta)(\cos^2\theta + \cos\theta\sin\theta + \sin^2\theta) \]
Since \( \cos^2\theta + \sin^2\theta = 1 \), we have:
\[ \cos^3\theta - \sin^3\theta = (\cos\theta - \sin\theta)(1 + \cos\theta\sin\theta) \]
Thus, the expression can be rewritten as:
\[ 4(\cos\theta - \sin\theta)(1 + \cos\theta\sin\theta) + 4\sin\theta\cos\theta - 6\sin\theta + 3\cos\theta \]

### Step 4: Analyze the positivity of the expression
For the expression to be positive, we need to analyze the critical points where \( \cos\theta - \sin\theta = 0 \). This occurs when:
\[ \tan\theta = 1 \quad \Rightarrow \quad \theta = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]

### Step 5: Determine the intervals
Now, we need to check the intervals around \( \theta = \frac{\pi}{4} \):
- For \( \theta < \frac{\pi}{4} \), \( \cos\theta > \sin\theta \) and the expression is positive.
- For \( \theta > \frac{\pi}{4} \), \( \sin\theta > \cos\theta \) and we need to check if the expression remains positive.

### Conclusion
The expression is positive when \( \cos\theta - \sin\theta > 0 \), which occurs in the intervals:
\[ \theta \in \left(2n\pi, 2n\pi + \frac{\pi}{4}\right) \cup \left(2n\pi + \frac{3\pi}{4}, 2(n+1)\pi\right), \quad n \in \mathbb{Z} \]