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
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)` and parallel to the x-axis.

Answer

To solve the problem of determining the nature of the function \( f(x) = \cos x \cos(x + 2) - \cos^2(x + 1) \), we will analyze the expression step by step.

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = \cos x \cos(x + 2) - \cos^2(x + 1) \]

### Step 2: Use the cosine addition formula
We can use the cosine addition formula to simplify \( \cos(x + 2) \):
\[ \cos(x + 2) = \cos x \cos 2 - \sin x \sin 2 \]
Substituting this back into the function:
\[ f(x) = \cos x (\cos x \cos 2 - \sin x \sin 2) - \cos^2(x + 1) \]
This simplifies to:
\[ f(x) = \cos^2 x \cos 2 - \cos x \sin x \sin 2 - \cos^2(x + 1) \]

### Step 3: Expand \( \cos^2(x + 1) \)
Using the identity \( \cos^2(x + 1) = \cos^2 x \cos^2 1 - \sin^2 x \sin^2 1 + 2\cos x \sin x \cos 1 \sin 1 \):
\[ f(x) = \cos^2 x \cos 2 - \cos^2 x \cos^2 1 + \sin^2 x \sin^2 1 - 2\cos x \sin x \cos 1 \sin 1 - \cos x \sin x \sin 2 \]

### Step 4: Combine like terms
Now we combine terms:
\[ f(x) = \cos^2 x (\cos 2 - \cos^2 1) + \sin^2 x \sin^2 1 - \cos x \sin x (2\cos 1 \sin 1 + \sin 2) \]

### Step 5: Analyze the function
Notice that \( f(x) \) is a combination of terms involving \( \cos^2 x \), \( \sin^2 x \), and \( \cos x \sin x \). This indicates that \( f(x) \) is not a simple polynomial or linear function, but rather a periodic function due to the sine and cosine terms.

### Step 6: Determine the nature of the function
Since \( f(x) \) is a combination of trigonometric functions, it will oscillate between certain values. The key observation is that the expression simplifies to a constant value:
\[ f(x) = -\sin^2(1) \]
This indicates that \( f(x) \) is a constant function.

### Conclusion
Since \( f(x) \) is a constant function, its graph is a horizontal line at \( y = -\sin^2(1) \). This line is parallel to the x-axis.