`Show that: sqrt(2+sqrt(2+sqrt(2+2cos8theta)))=2costheta,0 `

Explanation

To show that \( \sqrt{2 + \sqrt{2 + \sqrt{2 + 2 \cos 8\theta}}} = 2 \cos \theta \) for \( 0 < \theta < \frac{\pi}{8} \), we will simplify the left-hand side step by step.

### Step 1: Start with the innermost expression
We begin with the innermost part of the expression, which is \( 2 + 2 \cos 8\theta \).

Using the cosine double angle identity:
\[
\cos 8\theta = 2 \cos^2 4\theta - 1
...