Find the Value of `\ sqrt(2+sqrt(2+2cos4theta))`

Options

A. `2costheta`
B. `costheta`
C. `2cos(theta/2)`
D. `2cos(theta/4)`

Explanation

To find the value of \( \sqrt{2 + \sqrt{2 + 2 \cos 4\theta}} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{2 + \sqrt{2 + 2 \cos 4\theta}} \]

### Step 2: Factor out the common term
Notice that we can factor out 2 from the inner square root:
\[ \sqrt{2 + \sqrt{2(1 + \cos 4\theta)}} \]

### Step 3: Use the trigonometric identity
We know from trigonometric identities that:
\[ 1 + \cos 4\theta = 2 \cos^2 2\theta \]
Substituting this into our expression gives:
\[ \sqrt{2 + \sqrt{2 \cdot 2 \cos^2 2\theta}} = \sqrt{2 + \sqrt{4 \cos^2 2\theta}} \]

### Step 4: Simplify the inner square root
The inner square root simplifies to:
\[ \sqrt{4 \cos^2 2\theta} = 2 \cos 2\theta \]
Thus, our expression now looks like:
\[ \sqrt{2 + 2 \cos 2\theta} \]

### Step 5: Factor out the common term again
We can factor out 2 from the expression:
\[ \sqrt{2(1 + \cos 2\theta)} = \sqrt{2 \cdot 2 \cos^2 \theta} \quad \text{(using } 1 + \cos 2\theta = 2 \cos^2 \theta\text{)} \]

### Step 6: Simplify the expression
This simplifies to:
\[ \sqrt{4 \cos^2 \theta} = 2 \cos \theta \]

### Final Answer
Thus, the value of \( \sqrt{2 + \sqrt{2 + 2 \cos 4\theta}} \) is:
\[ \boxed{2 \cos \theta} \]