A factor of `cos 4theta - cos 4phi` is

Options

A. `cos theta + cos phi`
B. `cos theta + sin phi`
C. `cos theta - cos phi`
D. `cos theta - sin phi`

Answer

To find a factor of \( \cos 4\theta - \cos 4\phi \), we can use trigonometric identities. Let's break it down step by step.

### Step 1: Use the Cosine Difference Identity
We start with the expression:
\[ \cos 4\theta - \cos 4\phi \]
Using the cosine difference identity, we can express this as:
\[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
where \( A = 4\theta \) and \( B = 4\phi \). Therefore, we have:
\[ \cos 4\theta - \cos 4\phi = -2 \sin\left(\frac{4\theta + 4\phi}{2}\right) \sin\left(\frac{4\theta - 4\phi}{2}\right) \]

### Step 2: Simplify the Expression
Now, simplify the angles:
\[ \frac{4\theta + 4\phi}{2} = 2\theta + 2\phi \]
\[ \frac{4\theta - 4\phi}{2} = 2\theta - 2\phi \]
Thus, we can rewrite our expression as:
\[ \cos 4\theta - \cos 4\phi = -2 \sin(2\theta + 2\phi) \sin(2\theta - 2\phi) \]

### Step 3: Identify the Factors
From the expression we derived:
\[ \cos 4\theta - \cos 4\phi = -2 \sin(2\theta + 2\phi) \sin(2\theta - 2\phi) \]
We can see that the factors of \( \cos 4\theta - \cos 4\phi \) include:
1. \( \sin(2\theta + 2\phi) \)
2. \( \sin(2\theta - 2\phi) \)

### Conclusion
Thus, a factor of \( \cos 4\theta - \cos 4\phi \) is:
\[ \sin(2\theta + 2\phi) \quad \text{or} \quad \sin(2\theta - 2\phi) \]