You wrote it incorrectly, denominator is sin3x-cosx

Thank you for the clarification. The correct inequality from the image is:

\[
\frac{\sin(3x) + 2\cos(x)}{\sin(3x) - \cos(x)} > 1
\]

with \( x \) in the interval \([0, \frac{\pi}{2}]\).

### Solution:

1. **Rewrite the Inequality**:
   Start by subtracting 1 from both sides:
   \[
   \frac{\sin(3x) + 2\cos(x)}{\sin(3x) - \cos(x)} - 1 > 0
   \]

2. **Combine the Fractions**:
   Combine the terms over a common denominator:
   \[
   \frac{\sin(3x) + 2\cos(x) - (\sin(3x) - \cos(x))}{\sin(3x) - \cos(x)} > 0
   \]

   Simplify the numerator:
   \[
   \frac{\sin(3x) + 2\cos(x) - \sin(3x) + \cos(x)}{\sin(3x) - \cos(x)} > 0
   \]
   \[
   \frac{3\cos(x)}{\sin(3x) - \cos(x)} > 0
   \]

3. **Sign Analysis**:
   To solve the inequality, analyze the signs of the numerator and denominator separately:
   
   - **Numerator**: \( 3\cos(x) \) is positive when \( \cos(x) > 0 \). This is true for \( x \in \left(0, \frac{\pi}{2}\right) \).
   - **Denominator**: \( \sin(3x) - \cos(x) \).

   Determine where \( \sin(3x) - \cos(x) > 0 \) and where it is negative within the interval \([0, \frac{\pi}{2}]\).

4. **Critical Points**:
   Find the values of \( x \) where \( \sin(3x) = \cos(x) \) since these are the points where the denominator changes sign.

   The equation \( \sin(3x) = \cos(x) \) can be solved by using the identity \( \sin(3x) = \sin\left(2x + x\right) = 3\sin(x) - 4\sin^3(x) \), and setting it equal to \( \cos(x) \), then solving the resulting equation.

5. **Determine the Intervals**:
   Once you have the critical points, determine on which intervals the expression \( \frac{3\cos(x)}{\sin(3x) - \cos(x)} \) is positive.

6. **Final Solution**:
   The inequality holds where both the numerator and denominator are either both positive or both negative.

Would you like a detailed analysis on how to solve \( \sin(3x) = \cos(x) \) or a step-by-step walkthrough of solving this inequality?

### Follow-Up Questions:
1. How can we determine the sign of trigonometric expressions on specific intervals?
2. What methods can we use to solve the equation \( \sin(3x) = \cos(x) \)?
3. Why is it important to find the critical points when solving inequalities?
4. How do trigonometric identities help simplify complex expressions?
5. Can you always divide by a trigonometric function when solving inequalities?

### Tip:
When solving trigonometric inequalities, always consider the periodicity and symmetry of the functions involved to ensure you correctly analyze all intervals.

Explore how to solve the trigonometric inequality involving sin(3x) and cos(x) with a step-by-step explanation. This problem focuses on trigonometric identities and inequalities within the interval [0, π/2]. Suitable for advanced high school students and beyond.

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