If `y=(sin^(4)x-cos^(4)x+sin^(2) x cos^(2)x)/(sin^(4) x+ cos^(4)x + sin^(2) x cos^(2)x), x in (0, pi/2)`, then

Options

A. `-(3)/(2) le y le 1/2`
B. ` 1 le y le 1/2`
C. `-(5)/(3) le y le 1`
D. None of these

Answer

To solve the given problem, we need to simplify the expression for \( y \):

\[ y = \frac{\sin^4 x - \cos^4 x + \sin^2 x \cos^2 x}{\sin^4 x + \cos^4 x + \sin^2 x \cos^2 x} \]

### Step 1: Use the identity for \( a^4 - b^4 \)

Recall that \( a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \). We can apply this to \( \sin^4 x - \cos^4 x \):

\[ \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) \]

Since \( \sin^2 x + \cos^2 x = 1 \), we have:

\[ \sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x \]

### Step 2: Substitute back into the expression for \( y \)

Now substitute this back into the expression for \( y \):

\[ y = \frac{\sin^2 x - \cos^2 x + \sin^2 x \cos^2 x}{\sin^4 x + \cos^4 x + \sin^2 x \cos^2 x} \]

### Step 3: Simplify the denominator

Next, we simplify the denominator \( \sin^4 x + \cos^4 x \). We can use the identity:

\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \]

Thus, the denominator becomes:

\[ \sin^4 x + \cos^4 x + \sin^2 x \cos^2 x = (1 - 2\sin^2 x \cos^2 x) + \sin^2 x \cos^2 x = 1 - \sin^2 x \cos^2 x \]

### Step 4: Substitute the denominator back into \( y \)

Now substitute this back into the expression for \( y \):

\[ y = \frac{\sin^2 x - \cos^2 x + \sin^2 x \cos^2 x}{1 - \sin^2 x \cos^2 x} \]

### Step 5: Factor the numerator

The numerator can be rearranged:

\[ y = \frac{(\sin^2 x - \cos^2 x) + \sin^2 x \cos^2 x}{1 - \sin^2 x \cos^2 x} \]

### Step 6: Final simplification

At this point, we can evaluate \( y \) for specific values of \( x \) in the interval \( (0, \frac{\pi}{2}) \) or analyze its behavior. However, we can also see that:

\[ \sin^2 x - \cos^2 x = \sin^2 x - (1 - \sin^2 x) = 2\sin^2 x - 1 \]

Thus, we can write:

\[ y = \frac{(2\sin^2 x - 1) + \sin^2 x \cos^2 x}{1 - \sin^2 x \cos^2 x} \]

This expression can be evaluated further or analyzed depending on the context of the problem.