First slide

Multiple and sub- multiple Angles

Question

If $cosβ$ is the geometric mean between $\sin α and \cos α$ where $0 < α, β < π / 2$ then $\cos 2 β$ is equal to

Moderate

A

$- 2 \sin^{2} (\frac{π}{4} - α)$
B

$- 2 \cos^{2} (\frac{π}{4} + α)$
C

$2 \sin^{2} (\frac{π}{4} + α)$
D

$2 \cos^{2} (\frac{π}{4} - α)$

Solution

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

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