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COMEDK · Maths · 8. Trigonometric Ratios & Identities

\(\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 \theta}}} \text { where } \theta \in\left[-\dfrac{\pi}{8}, \dfrac{\pi}{8}\right] \text { is equal to }\)

  1. A \(\cos 2 \theta\)
  2. B \(\sin 2 \theta\)
  3. C \(2 \sin \theta\)
  4. D \(2 \cos \theta\)
Verified Solution

Answer & Solution

Correct Answer

(D) \(2 \cos \theta\)

Step-by-step Solution

Detailed explanation

Let the given expression be \(E = \sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 \theta}}}\). Using the trigonometric identity \(1 + \cos 2A = 2 \cos^2 A\), we have \(2 + 2 \cos 8 \theta = 2(1 + \cos 8 \theta) = 2(2 \cos^2 4 \theta) = 4 \cos^2 4 \theta\). Since…