Question

The value of \( \sqrt{2 + \sqrt{2 + 2 \cos 4\theta}} \) is

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

Hint:

The assumptions \( \cos 2\theta \ge 0 \) and \( \cos \theta \ge 0 \) are crucial here. The value of the expression depends on the range of \( \theta \). However, given the options, the simplified form \( 2 \cos \theta \) is the most likely intended answer, implying these assumptions hold for the context of the problem.

Answer 1

The Correct Option is B

Step 1: Simplify the innermost term using the double angle formula.
We use the trigonometric identity \( 1 + \cos 2A = 2 \cos^2 A \). The innermost part of the expression involves \( 2 + 2 \cos 4\theta \):
$$2 + 2 \cos 4\theta = 2(1 + \cos 4\theta)$$ Here, \( 2A = 4\theta \), so \( A = 2\theta \). Applying the identity:
$$2(1 + \cos 4\theta) = 2(2 \cos^2 2\theta) = 4 \cos^2 2\theta$$ Step 2: Substitute the simplified term back into the expression and take the square root.
The expression now becomes \( \sqrt{2 + \sqrt{4 \cos^2 2\theta}} \).
$$\sqrt{4 \cos^2 2\theta} = |2 \cos 2\theta|$$ Assuming \( \cos 2\theta \ge 0 \), this simplifies to \( 2 \cos 2\theta \). The expression is now:
$$\sqrt{2 + 2 \cos 2\theta}$$ Step 3: Simplify the remaining expression using the double angle formula again.
We have \( \sqrt{2 + 2 \cos 2\theta} = \sqrt{2(1 + \cos 2\theta)} \). Here, \( 2A = 2\theta \), so \( A = \theta \). Applying the identity \( 1 + \cos 2A = 2 \cos^2 A \):
$$\sqrt{2(1 + \cos 2\theta)} = \sqrt{2(2 \cos^2 \theta)} = \sqrt{4 \cos^2 \theta}$$
Step 4: Take the final square root.
$$\sqrt{4 \cos^2 \theta} = |2 \cos \theta|$$ Assuming \( \cos \theta \ge 0 \), the final value is \( 2 \cos \theta \).