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The principal solution set of the equation, 2 cos x = sqrt(2+2sin2x)is

$2 \cos x = \sqrt{2 + 2 \sin 2 x} \Rightarrow 2 \cos x = \sqrt{2 (1 + 2 \sin x \cos x)} \Rightarrow 2 \cos x = \sqrt{2 (\sin x + \cos x)^{2}} = \sqrt{2} (\sin x + \cos x) \Rightarrow \cos x = \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \Rightarrow \cos x = \cos (\frac{π}{4} - x) \Rightarrow 2 x = \frac{π}{4} \Rightarrow x = \frac{π}{8}$

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