Solve `sqrt(2+sqrt(2+sqrt(2+2cos16theta)))`

Options

A. `2cos2theta`
B. `sqrt(2)costheta`
C. `2sintheta`
D. `sqrt(2)sintheta`

Answer

To solve the expression \( \sqrt{2 + \sqrt{2 + \sqrt{2 + 2 \cos 16\theta}}} \), we can follow these steps:

### Step 1: Simplify the innermost expression
Start with the innermost expression \( 2 + 2 \cos 16\theta \).

Using the identity \( 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \), we can rewrite \( 2 + 2 \cos 16\theta \) as:
\[ 2(1 + \cos 16\theta) = 2 \cdot 2 \cos^2(8\theta) = 4 \cos^2(8\theta) \]

### Step 2: Substitute back into the expression
Now, substitute this back into the expression:
\[ \sqrt{2 + \sqrt{2 + 4 \cos^2(8\theta)}} \]

### Step 3: Simplify the next layer
Now simplify \( 2 + 4 \cos^2(8\theta) \):
\[ 2 + 4 \cos^2(8\theta) = 2(1 + 2 \cos^2(8\theta)) = 2(1 + \cos 16\theta) = 4 \cos^2(4\theta) \]

### Step 4: Substitute again
Now substitute this back into the expression:
\[ \sqrt{2 + \sqrt{4 \cos^2(4\theta)}} \]

### Step 5: Simplify the square root
Now simplify \( \sqrt{4 \cos^2(4\theta)} \):
\[ \sqrt{4 \cos^2(4\theta)} = 2 \cos(4\theta) \]

### Step 6: Substitute one last time
Now substitute this back into the expression:
\[ \sqrt{2 + 2 \cos(4\theta)} \]

### Step 7: Final simplification
Using the identity again \( 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \):
\[ 2 + 2 \cos(4\theta) = 2(1 + \cos(4\theta)) = 4 \cos^2(2\theta) \]

### Step 8: Final expression
Now take the square root:
\[ \sqrt{4 \cos^2(2\theta)} = 2 \cos(2\theta) \]

Thus, the final answer is:
\[ \sqrt{2 + \sqrt{2 + \sqrt{2 + 2 \cos 16\theta}}} = 2 \cos(2\theta) \]

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