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Let $x, y, z \in (0, \frac{π}{2})$ are first three consecutive terms of an arithmetic progression such that $\cos x + \cos y + \cos z = 1$ and $\sin x + \sin y + \sin z = \frac{1}{\sqrt{2}},$ then which of the following is/are correct?

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$\tan y = \sqrt{2}$

$\cos (x - y) = \frac{\sqrt{3} - \sqrt{2}}{2 \sqrt{2}}$

$\tan 2 y = \frac{2 \sqrt{2}}{3}$

$\sin (x - y) + \sin (y - z) = 0$

answer is B.

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Detailed Solution

Let $x = y - θ, z = y + θ$

$\Rightarrow \cos (y - θ) + \cos y + \cos (y + θ) = 1 = \frac{\sin \frac{3 θ}{2}}{\sin \frac{θ}{2}} . \cos (y)$

and $\sin (y - θ) + \sin y + \sin (y + θ) = \frac{1}{\sqrt{2}} = \frac{\sin \frac{3 θ}{2}}{\sin \frac{θ}{2}} . \sin (y)$

$\Rightarrow \cos θ = \frac{\sqrt{3} - \sqrt{2}}{2 \sqrt{2}}$

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