The range of `y=(sin4x-sin2x)/(sin4x+sin2x)` satisfies

Options

A. `y in (-oo, (1)/(3))`
B. `y in ((1)/(3),1)`
C. `y in (1, 3)`
D. `y in (3, oo)`

Answer

To find the range of the function \( y = \frac{\sin 4x - \sin 2x}{\sin 4x + \sin 2x} \), we can follow these steps:

### Step 1: Use the sine subtraction and addition formulas
We know that:
\[ \sin c - \sin d = 2 \cos\left(\frac{c + d}{2}\right) \sin\left(\frac{c - d}{2}\right) \]
\[ \sin c + \sin d = 2 \sin\left(\frac{c + d}{2}\right) \cos\left(\frac{c - d}{2}\right) \]
Here, let \( c = 4x \) and \( d = 2x \).

### Step 2: Substitute into the formulas
Using the above formulas:
\[ \sin 4x - \sin 2x = 2 \cos\left(\frac{4x + 2x}{2}\right) \sin\left(\frac{4x - 2x}{2}\right) = 2 \cos(3x) \sin(x) \]
\[ \sin 4x + \sin 2x = 2 \sin\left(\frac{4x + 2x}{2}\right) \cos\left(\frac{4x - 2x}{2}\right) = 2 \sin(3x) \cos(x) \]

### Step 3: Rewrite the function
Now substituting these back into the function \( y \):
\[ y = \frac{2 \cos(3x) \sin(x)}{2 \sin(3x) \cos(x)} = \frac{\cos(3x) \sin(x)}{\sin(3x) \cos(x)} \]
This simplifies to:
\[ y = \tan(3x) \cdot \frac{\sin(x)}{\cos(x)} = \tan(3x) \tan(x) \]

### Step 4: Set \( t = \tan(x) \)
Let \( t = \tan(x) \). Then:
\[ y = \tan(3x) \cdot t \]

### Step 5: Use the tangent triple angle formula
The formula for \( \tan(3x) \) is:
\[ \tan(3x) = \frac{3t - t^3}{1 - 3t^2} \]
Substituting this into our equation for \( y \):
\[ y = \frac{3t - t^3}{1 - 3t^2} \cdot t = \frac{3t^2 - t^4}{1 - 3t^2} \]

### Step 6: Analyze the range of \( y \)
To find the range of \( y \), we need to analyze the expression:
\[ y = \frac{3t^2 - t^4}{1 - 3t^2} \]
We can set \( t^2 = u \) (where \( u \geq 0 \)):
\[ y = \frac{3u - u^2}{1 - 3u} \]

### Step 7: Find critical points and intervals
To find the range, we need to find where the function is defined and where it becomes zero or undefined:
1. The denominator \( 1 - 3u \) must not be zero, so \( u \neq \frac{1}{3} \).
2. The numerator \( 3u - u^2 = 0 \) gives \( u(3 - u) = 0 \) leading to \( u = 0 \) or \( u = 3 \).

### Step 8: Determine the intervals
The critical points are \( u = 0 \), \( u = \frac{1}{3} \), and \( u = 3 \). We can test intervals around these points to determine the sign of \( y \):
- For \( u < 0 \): Not applicable as \( u \geq 0 \).
- For \( 0 < u < \frac{1}{3} \): \( y > 0 \).
- For \( u = \frac{1}{3} \): Undefined.
- For \( \frac{1}{3} < u < 3 \): \( y < 0 \).
- For \( u = 3 \): \( y = 0 \).
- For \( u > 3 \): \( y \to -\infty \).

### Step 9: Conclusion
The range of \( y \) is:
\[ (-\infty, 0) \cup \left(0, \frac{1}{3}\right) \cup (3, \infty) \]