The equation `y=sinxsin(x+2)-sin^(2)(x+1)` represents a straight line lying in

Options

A. second and third quadrants only
B. first, third and fourth quadrants
C. first, second and fourth quadrants
D. third and fourth quadrants only

Explanation

To solve the equation \( y = \sin x \sin(x + 2) - \sin^2(x + 1) \) and determine the quadrants in which it lies, we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We start with the equation:
\[ y = \sin x \sin(x + 2) - \sin^2(x + 1) \]
Using the product-to-sum identities, we can rewrite \( \sin x \sin(x + 2) \):
\[ \sin x \sin(x + 2) = \frac{1}{2} [\cos(x - (x + 2)) - \cos(x + (x + 2))] = \frac{1}{2} [\cos(-2) - \cos(2x + 2)] \]
Thus, we have:
\[ y = \frac{1}{2} [\cos(2) - \cos(2x + 2)] - \sin^2(x + 1) \]

### Step 2: Simplify the equation
Next, we simplify \( \sin^2(x + 1) \) using the identity \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \):
\[ \sin^2(x + 1) = \frac{1 - \cos(2(x + 1))}{2} = \frac{1 - \cos(2x + 2)}{2} \]
Substituting this back into the equation gives:
\[ y = \frac{1}{2} [\cos(2) - \cos(2x + 2)] - \frac{1 - \cos(2x + 2)}{2} \]

### Step 3: Combine like terms
Combining the terms, we have:
\[ y = \frac{1}{2} \cos(2) - \frac{1}{2} + \frac{1}{2} \cos(2x + 2) - \frac{1}{2} \cos(2x + 2) \]
This simplifies to:
\[ y = \frac{1}{2} \cos(2) - \frac{1}{2} \]
This indicates that \( y \) is a constant value.

### Step 4: Determine the sign of \( y \)
Now, we need to evaluate the constant:
\[ y = \frac{1}{2} (\cos(2) - 1) \]
Since \( \cos(2) \) is approximately \( -0.416 \), we find:
\[ y = \frac{1}{2} (-0.416 - 1) = \frac{1}{2} (-1.416) < 0 \]
Thus, \( y \) is always negative.

### Step 5: Identify the quadrants
Since \( y \) is always negative, the line represented by the equation lies in the third and fourth quadrants.

### Conclusion
The equation \( y = \sin x \sin(x + 2) - \sin^2(x + 1) \) represents a straight line lying in the **third and fourth quadrants**.