Find the set of values of x , which satisfy `sin x * cos^(3) x gt cos x* sin^(3) x , 0 le x le 2pi` .

Options

A. `x in (0, pi/4) uu ( pi/2, (3pi)/(4))`
B. `x in (pi/4, pi/2) uu ((3pi)/(4), pi)`
C. `x in (pi/4, pi/2)`
D. None of these

Answer

To solve the inequality \( \sin x \cdot \cos^3 x > \cos x \cdot \sin^3 x \) for \( 0 \leq x \leq 2\pi \), we can follow these steps:

### Step 1: Rearranging the Inequality
We start with the inequality:
\[ \sin x \cdot \cos^3 x - \cos x \cdot \sin^3 x > 0 \]
This can be rewritten as:
\[ \sin x \cdot \cos^3 x - \sin^3 x \cdot \cos x > 0 \]

### Step 2: Factoring the Left Side
We can factor out \( \sin x \cdot \cos x \):
\[ \sin x \cdot \cos x \left( \cos^2 x - \sin^2 x \right) > 0 \]

### Step 3: Using Trigonometric Identities
Using the identity \( \cos^2 x - \sin^2 x = \cos(2x) \), we can rewrite the inequality as:
\[ \sin x \cdot \cos x \cdot \cos(2x) > 0 \]

### Step 4: Analyzing the Factors
Now we need to analyze when each factor is positive:
1. \( \sin x > 0 \)
2. \( \cos x > 0 \)
3. \( \cos(2x) > 0 \)

### Step 5: Finding Intervals for Each Factor
1. **For \( \sin x > 0 \)**: This occurs in the intervals:
\[ (0, \pi) \]
2. **For \( \cos x > 0 \)**: This occurs in the intervals:
\[ (0, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi) \]
3. **For \( \cos(2x) > 0 \)**: This occurs in the intervals:
\[ (2n\pi - \frac{\pi}{2}, 2n\pi + \frac{\pi}{2}) \quad \text{for } n \in \mathbb{Z} \]
For \( 0 \leq 2x \leq 4\pi \), we find:
\[ (0, \frac{\pi}{2}) \cup (2\pi, \frac{5\pi}{2}) \]
Dividing by 2 gives:
\[ (0, \frac{\pi}{4}) \cup ( \pi, \frac{5\pi}{4}) \]

### Step 6: Finding the Intersection of Intervals
Now we find the intersection of the intervals:
- From \( \sin x > 0 \): \( (0, \pi) \)
- From \( \cos x > 0 \): \( (0, \frac{\pi}{2}) \)
- From \( \cos(2x) > 0 \): \( (0, \frac{\pi}{4}) \) and \( (\pi, \frac{5\pi}{4}) \)

The valid intervals where all three conditions are satisfied are:
- From \( (0, \frac{\pi}{4}) \)
- From \( (\pi, \frac{5\pi}{4}) \) does not intersect with \( (0, \pi) \)

### Final Solution
Thus, the set of values of \( x \) that satisfy the inequality is:
\[ x \in \left(0, \frac{\pi}{4}\right) \]