Let ` f( theta) = ( cos theta - "cos" (pi)/(8))(cos theta - "cos" (3 pi)/(8))(cos theta - "cos" (5 pi)/(8) )(cos theta - "cos" (7pi)/(8))` then :

Options

A. maximum value of ` f(theta ) AA theta in R ` is ` (1)/(4) `
B. maximum value of ` f(theta) AA theta in R ` is ` (1)/(8) `
C. ` f(0)= (1)/(8) `
D. Number of principle solutions of ` f(theta) = 0 ` is 8

Answer

To solve the problem, we need to analyze the function \( f(\theta) = \left( \cos \theta - \cos \frac{\pi}{8} \right) \left( \cos \theta - \cos \frac{3\pi}{8} \right) \left( \cos \theta - \cos \frac{5\pi}{8} \right) \left( \cos \theta - \cos \frac{7\pi}{8} \right) \).

### Step 1: Rewrite the cosines
We can rewrite \( \cos \frac{5\pi}{8} \) and \( \cos \frac{7\pi}{8} \) using the identity \( \cos(\pi - x) = -\cos(x) \):
- \( \cos \frac{5\pi}{8} = \cos\left(\pi - \frac{3\pi}{8}\right) = -\cos \frac{3\pi}{8} \)
- \( \cos \frac{7\pi}{8} = \cos\left(\pi - \frac{\pi}{8}\right) = -\cos \frac{\pi}{8} \)

Thus, we can express \( f(\theta) \) as:
\[ f(\theta) = \left( \cos \theta - \cos \frac{\pi}{8} \right) \left( \cos \theta - \cos \frac{3\pi}{8} \right) \left( \cos \theta + \cos \frac{3\pi}{8} \right) \left( \cos \theta + \cos \frac{\pi}{8} \right) \]

### Step 2: Use the identity for products
Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we can simplify:
\[ f(\theta) = \left( \cos^2 \theta - \cos^2 \frac{\pi}{8} \right) \left( \cos^2 \theta - \cos^2 \frac{3\pi}{8} \right) \]

### Step 3: Substitute \( \cos^2 \frac{3\pi}{8} \)
We know that \( \cos \frac{3\pi}{8} = \sin \frac{\pi}{8} \), so:
\[ \cos^2 \frac{3\pi}{8} = \sin^2 \frac{\pi}{8} \]

Thus, we can rewrite \( f(\theta) \) as:
\[ f(\theta) = \left( \cos^2 \theta - \cos^2 \frac{\pi}{8} \right) \left( \cos^2 \theta - \sin^2 \frac{\pi}{8} \right) \]

### Step 4: Expand the product
Now, we expand this product:
\[ f(\theta) = \cos^4 \theta - \cos^2 \frac{\pi}{8} \cos^2 \theta - \sin^2 \frac{\pi}{8} \cos^2 \theta + \cos^2 \frac{\pi}{8} \sin^2 \frac{\pi}{8} \]
\[ = \cos^4 \theta - \cos^2 \theta \left( \cos^2 \frac{\pi}{8} + \sin^2 \frac{\pi}{8} \right) + \cos^2 \frac{\pi}{8} \sin^2 \frac{\pi}{8} \]

Using the identity \( \cos^2 \frac{\pi}{8} + \sin^2 \frac{\pi}{8} = 1 \):
\[ f(\theta) = \cos^4 \theta - \cos^2 \theta + \cos^2 \frac{\pi}{8} \sin^2 \frac{\pi}{8} \]

### Step 5: Factor out \( \cos^2 \theta \)
Let \( x = \cos^2 \theta \):
\[ f(\theta) = x^2 - x + \cos^2 \frac{\pi}{8} \sin^2 \frac{\pi}{8} \]

### Step 6: Find the maximum value
The maximum value occurs when \( x = \frac{1}{2} \) (vertex of the quadratic):
\[ f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + \cos^2 \frac{\pi}{8} \sin^2 \frac{\pi}{8} \]

### Step 7: Calculate \( f(0) \)
Substituting \( \theta = 0 \):
\[ f(0) = \left( 1 - \cos \frac{\pi}{8} \right) \left( 1 - \cos \frac{3\pi}{8} \right) \left( 1 + \cos \frac{3\pi}{8} \right) \left( 1 + \cos \frac{\pi}{8} \right) \]
This will yield \( f(0) = \frac{1}{8} \).

### Step 8: Find the number of solutions for \( f(\theta) = 0 \)
Setting \( f(\theta) = 0 \) leads to solving:
\[ \cos^4 \theta - \cos^2 \theta + \cos^2 \frac{\pi}{8} \sin^2 \frac{\pi}{8} = 0 \]
This will give us the number of principal solutions.