If $\sin β$ is the GM between $\sin α and \cos α,$ then $\cos 2 β$ is equal to

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$2 \cos^{2} (\frac{π}{4} - α)$

$2 \cos^{2} (\frac{3 π}{4} + 2 α)$

$2 \sin^{2} (\frac{π}{4} - α)$

$2 \sin^{2} (\frac{π}{4} + α)$

answer is A.

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Detailed Solution

Since $\sin β$ is the GM between $\sin a a n d \cos a .$

$\begin{array}{l} ∴ \sin^{2} β = \sin α \cos α \\ \Rightarrow \cos 2 β = 1 - 2 \sin^{2} β = 1 - 2 \sin^{2} α \cos^{2} α \\ \Rightarrow \cos 2 β = (\cos α - \sin α)^{2} \\ \Rightarrow \cos 2 β = 2 {\{\frac{1}{\sqrt{2}} \cos α - \frac{1}{\sqrt{2}} \sin α\}}^{2} \\ \Rightarrow \cos 2 β = 2 \cos^{2} (\frac{π}{4} + α) = 2 \sin^{2} (\frac{π}{4} - α) \end{array}$