$\sqrt{2+\sqrt{2+2 \cos 4 \theta}}=$

$\sqrt{2+\sqrt{2+2 \cos 4 \theta}}=$
  1. $2 \cos \theta$
  2. $\frac{\cos \theta}{2}$
  3. $\frac{\cos \theta}{\sqrt{2}}$
  4. $\sqrt{2} \cdot \cos \theta$

Solution

$\begin{aligned} \sqrt{2+\sqrt{2+2 \cos 4 \theta}} &=\sqrt{2+\sqrt{2(1+\cos 4 \theta)}} \\ &=\sqrt{2+\sqrt{2 \times 2 \cos ^{2} 2 \theta}}=\sqrt{2+2 \cos ^{2} 2 \theta} \\ &=\sqrt{2\left(1+\cos ^{2} 2 \theta\right)}=\sqrt{2 \times 2 \cos ^{2} \theta}=2 \cos \theta \end{aligned}$

Asked in: MHT CET 2020 (15 Oct Shift 2)

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