Question

\(\sqrt{2 + \sqrt{2 + \sqrt{2 + 2\cos(8\theta)}}}\)=

• A. \(\sin 2\theta\)
• B. \(2\sin \theta\)
• C. \(2\cos \theta\)
• D. \(2 \cos \frac{\theta}{2}\)

Answer 1

The Correct Option is C

Let's simplify the expression step by step:

Let the expression be: \[ \sqrt{2 + \sqrt{2 + \sqrt{2 + 2\cos(8\theta)}}} \]

We use the identity: \[ \cos(2\theta) = 2\cos^2\theta - 1 \Rightarrow \cos^2\theta = \frac{1 + \cos(2\theta)}{2} \]
Also, iteratively, \[ \cos(4\theta) = 2\cos^2(2\theta) - 1,\quad \cos(8\theta) = 2\cos^2(4\theta) - 1 \]

From trigonometric nested root identities, \[ \sqrt{2 + \sqrt{2 + \sqrt{2 + 2\cos(8\theta)}}} = 2\cos\theta \]

This identity is derived using the recursive form of cosine angle reduction.

Correct Answer: \(2\cos\theta\)

Answer 2

We want to simplify the expression \(\sqrt{2 + \sqrt{2 + \sqrt{2 + 2\cos(8\theta)}}}\). 

We will use the half-angle identity for cosine repeatedly: \(1 + \cos(2A) = 2\cos^2(A)\), which implies \(2 + 2\cos(2A) = 4\cos^2(A)\).

Let's start simplifying from the innermost square root:

\[ \sqrt{2 + 2\cos(8\theta)} \] Using the identity with \(2A = 8\theta\), so \(A = 4\theta\): \[ \sqrt{2(1 + \cos(8\theta))} = \sqrt{2(2\cos^2(4\theta))} = \sqrt{4\cos^2(4\theta)} \] Assuming \(0 \le 4\theta \le \frac{\pi}{2}\) so that \(\cos(4\theta) \ge 0\), this simplifies to: \[ = 2\cos(4\theta) \]

Now substitute this back into the expression:

\[ \sqrt{2 + \sqrt{2 + 2\cos(4\theta)}} \]

Simplify the inner square root again:

\[ \sqrt{2 + 2\cos(4\theta)} \] Using the identity with \(2A = 4\theta\), so \(A = 2\theta\): \[ \sqrt{2(1 + \cos(4\theta))} = \sqrt{2(2\cos^2(2\theta))} = \sqrt{4\cos^2(2\theta)} \] Assuming \(0 \le 2\theta \le \frac{\pi}{2}\) so that \(\cos(2\theta) \ge 0\), this simplifies to: \[ = 2\cos(2\theta) \]

Substitute this back into the expression:

\[ \sqrt{2 + 2\cos(2\theta)} \]

Simplify the final square root:

\[ \sqrt{2 + 2\cos(2\theta)} \] Using the identity with \(2A = 2\theta\), so \(A = \theta\): \[ \sqrt{2(1 + \cos(2\theta))} = \sqrt{2(2\cos^2(\theta))} = \sqrt{4\cos^2(\theta)} \] Assuming \(0 \le \theta \le \frac{\pi}{2}\) so that \(\cos(\theta) \ge 0\), this simplifies to: \[ = 2\cos(\theta) \]

Thus, under the assumption that the relevant angles are in the first quadrant where cosine is non-negative, the expression simplifies to \(2\cos(\theta)\).

Comparing this with the given options, the correct option is:

\(2\cos \theta\)