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

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 given inequality:
\[ \sin x \cdot \cos^3 x - \cos x \cdot \sin^3 x > 0 \]
This can be rearranged as:
\[ \sin x \cdot \cos^3 x - \cos x \cdot \sin^3 x = 0 \]

### Step 2: Factoring the Expression
Next, we can factor out common terms:
\[ \sin x \cdot \cos x \left( \cos^2 x - \sin^2 x \right) > 0 \]
This gives us two factors to consider:
1. \( \sin x \cdot \cos x \)
2. \( \cos^2 x - \sin^2 x \)

### Step 3: Using Trigonometric Identities
We know that:
\[ \sin x \cdot \cos x = \frac{1}{2} \sin(2x) \]
Thus, we can rewrite the inequality as:
\[ \frac{1}{2} \sin(2x) \left( \cos^2 x - \sin^2 x \right) > 0 \]
We can also express \( \cos^2 x - \sin^2 x \) as \( \cos(2x) \):
\[ \frac{1}{2} \sin(2x) \cdot \cos(2x) > 0 \]

### Step 4: Simplifying the Inequality
Multiplying both sides by 2 (which is positive) gives:
\[ \sin(2x) \cdot \cos(2x) > 0 \]

### Step 5: Analyzing the Sign of the Product
The product \( \sin(2x) \cdot \cos(2x) > 0 \) implies that both \( \sin(2x) \) and \( \cos(2x) \) must be either both positive or both negative.

### Step 6: Finding Intervals
1. **For \( \sin(2x) > 0 \)**:
- This occurs in the intervals \( (0, \pi) \) and \( (2\pi, 3\pi) \).

2. **For \( \cos(2x) > 0 \)**:
- This occurs in the intervals \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) and \( (\frac{3\pi}{2}, \frac{5\pi}{2}) \).

### Step 7: Finding Common Intervals
Now we need to find the common intervals where both conditions are satisfied:
- For \( 0 \leq 2x \leq 2\pi \):
- \( 0 < 2x < \pi \) gives \( 0 < x < \frac{\pi}{2} \)
- \( \frac{3\pi}{2} < 2x < 2\pi \) gives \( \frac{3\pi}{4} < x < \pi \)

Combining these intervals, we have:
- \( 0 < x < \frac{\pi}{4} \)
- \( \frac{\pi}{2} < x < \frac{3\pi}{4} \)
- \( \frac{3\pi}{2} < x < \frac{7\pi}{4} \)

### Step 8: Final Result
Thus, the solution set for \( x \) is:
\[ x \in \left(0, \frac{\pi}{4}\right) \cup \left(\frac{\pi}{2}, \frac{3\pi}{4}\right) \cup \left(\frac{3\pi}{2}, \frac{7\pi}{4}\right) \]