The graph of the function, `cos x cos (x +2) -cos^2 (x+1)` is

Options

A. a straight line passing through `(0-sin^(2)1)` with slope 2
B. a straight line passing through (0,0)
C. a parabola with vertex `(1-sin^(2)1)`
D. a straight line passing through the point `(pi/2,-sin^(2)1)` are parallel to the X-axis

Answer

To find the graph of the function \( y = \cos x \cos(x + 2) - \cos^2(x + 1) \), we will simplify the expression step by step and analyze its properties.

### Step 1: Write the function
Start with the given function:
\[ y = \cos x \cos(x + 2) - \cos^2(x + 1) \]

### Step 2: Use trigonometric identities
We can use the product-to-sum identities to simplify \( \cos x \cos(x + 2) \):
\[ \cos x \cos(x + 2) = \frac{1}{2} \left( \cos(2) + \cos(2x + 2) \right) \]
Thus, we can rewrite \( y \):
\[ y = \frac{1}{2} \left( \cos(2) + \cos(2x + 2) \right) - \cos^2(x + 1) \]

### Step 3: Expand \( \cos^2(x + 1) \)
Using the identity \( \cos^2 A = \frac{1 + \cos(2A)}{2} \), we can expand \( \cos^2(x + 1) \):
\[ \cos^2(x + 1) = \frac{1 + \cos(2(x + 1))}{2} = \frac{1 + \cos(2x + 2)}{2} \]
Substituting this back into our expression for \( y \):
\[ y = \frac{1}{2} \left( \cos(2) + \cos(2x + 2) \right) - \frac{1 + \cos(2x + 2)}{2} \]

### Step 4: Combine terms
Now, combine the terms:
\[ y = \frac{1}{2} \cos(2) + \frac{1}{2} \cos(2x + 2) - \frac{1}{2} - \frac{1}{2} \cos(2x + 2) \]
The \( \cos(2x + 2) \) terms cancel out:
\[ y = \frac{1}{2} \cos(2) - \frac{1}{2} \]

### Step 5: Simplify further
This can be simplified to:
\[ y = \frac{1}{2} (\cos(2) - 1) \]
Since \( \cos(2) \) is a constant, \( y \) is a constant value.

### Step 6: Conclusion about the graph
Since \( y \) is a constant, the graph of this function is a horizontal line. The specific value of \( y \) is \( \frac{1}{2} (\cos(2) - 1) \), which is a constant value.

### Final Answer
The graph of the function is a straight line parallel to the x-axis.