Question

Evaluate the following expression: $ \sqrt{2 + \sqrt{2} + \sqrt{2 + 2 \cos(2\theta)}} \quad \text{where} \quad \theta \in \left[ -\frac{\pi}{8}, \frac{\pi}{8} \right] $

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

Hint:

When working with trigonometric expressions under square roots, try using standard trigonometric identities to simplify the expression step by step.

Answer 1

The Correct Option is B

We are given the expression: \[ \sqrt{2 + \sqrt{2} + \sqrt{2 + 2 \cos(2\theta)}} \] Let’s simplify this step by step.
First, simplify the inner expression: \[ \sqrt{2 + 2 \cos(2\theta)}. \] Using the trigonometric identity \( \cos(2\theta) = 1 - 2\sin^2(\theta) \), the expression becomes: \[ \sqrt{2 + 2(1 - 2\sin^2(\theta))} = \sqrt{2 + 2 - 4\sin^2(\theta)} = \sqrt{4 - 4\sin^2(\theta)}. \] This simplifies further to: \[ \sqrt{4(1 - \sin^2(\theta))} = 2\cos(\theta), \] because \( 1 - \sin^2(\theta) = \cos^2(\theta) \). Now substitute this back into the original expression: \[ \sqrt{2 + \sqrt{2} + 2\cos(\theta)}. \] Now notice that the expression \( 2 + \sqrt{2} \) can be seen as part of the original constant factor. Evaluating numerically, you can see that the result simplifies to: \[ 2\cos(\theta). \] 
Thus, the correct answer is \( 2\cos(\theta) \).