Simplify Trigonometric Radical Expression

The goal is to simplify the expression $\sqrt{2 + \sqrt{2 + 2\cos 4\theta}}$. We will use trigonometric identities to simplify the nested square roots.

Step 1: Simplify the inner square root

Focus on the term $\sqrt{2 + 2\cos 4\theta}$.

• Factor out 2: $\sqrt{2(1 + \cos 4\theta)}$
• Use the double-angle identity $1 + \cos(2\alpha) = 2\cos^2 \alpha$. Here, $2\alpha = 4\theta$, so $\alpha = 2\theta$.
• Substitute the identity: $\sqrt{2(2\cos^2 2\theta)} = \sqrt{4\cos^2 2\theta}$
• Simplify the square root: $\sqrt{4\cos^2 2\theta} = |2\cos 2\theta|$

Assuming $\cos 2\theta \ge 0$, this simplifies to $2\cos 2\theta$.

Step 2: Substitute and simplify the outer square root

Substitute the result back into the original expression:

The expression becomes $\sqrt{2 + 2\cos 2\theta}$.

• Factor out 2: $\sqrt{2(1 + \cos 2\theta)}$
• Use the double-angle identity $1 + \cos(2\beta) = 2\cos^2 \beta$. Here, $2\beta = 2\theta$, so $\beta = \theta$.
• Substitute the identity: $\sqrt{2(2\cos^2 \theta)} = \sqrt{4\cos^2 \theta}$
• Simplify the square root: $\sqrt{4\cos^2 \theta} = |2\cos \theta|$

Step 3: Final Simplification

Assuming $\cos \theta \ge 0$, the expression simplifies to $2\cos \theta$.

Therefore, the simplified form of $\sqrt{2 + \sqrt{2 + 2\cos 4\theta}}$ is $2\cos \theta$. This corresponds to Option C.