`(3-4 cos 2theta + cos 4 theta)` equals

Options

A. `8cos 4 theta`
B. `8sin 4 theta`
C. `4sin^(4)theta`
D. `4cos^(4)theta`

Answer

To solve the expression \(3 - 4 \cos 2\theta + \cos 4\theta\), we will use trigonometric identities to simplify it step by step.

### Step 1: Rewrite \(\cos 4\theta\)
We can use the double angle formula for cosine, which states:
\[ \cos 4\theta = 2\cos^2 2\theta - 1 \]
So, we can substitute this into our expression:
\[ 3 - 4 \cos 2\theta + \cos 4\theta = 3 - 4 \cos 2\theta + (2\cos^2 2\theta - 1) \]

### Step 2: Combine like terms
Now, we can combine the constants:
\[ = (3 - 1) - 4 \cos 2\theta + 2\cos^2 2\theta = 2 - 4 \cos 2\theta + 2\cos^2 2\theta \]

### Step 3: Factor out the common term
Next, we can factor out a 2 from the expression:
\[ = 2(1 - 2 \cos 2\theta + \cos^2 2\theta) \]

### Step 4: Recognize a perfect square
The expression inside the parentheses can be recognized as a perfect square:
\[ 1 - 2 \cos 2\theta + \cos^2 2\theta = (1 - \cos 2\theta)^2 \]
Thus, we can rewrite our expression as:
\[ = 2(1 - \cos 2\theta)^2 \]

### Step 5: Use the identity for sine
We know that \(1 - \cos 2\theta = 2\sin^2 \theta\) (using the identity \(1 - \cos x = 2\sin^2 \frac{x}{2}\)):
\[ = 2(2\sin^2 \theta)^2 = 2 \cdot 4\sin^4 \theta = 8\sin^4 \theta \]

### Final Result
Thus, the expression simplifies to:
\[ 3 - 4 \cos 2\theta + \cos 4\theta = 8\sin^4 \theta \]