First slide

Multiple and sub- multiple Angles

Question

If $\cos α + \cos β = 0 = \sin α + \sin β,$ then $\cos 2 α + \cos 2 β =$

Moderate

A

$- 2 \sin (α + β)$
B

$- 2 \cos (α + β)$
C

$2 \sin (α + β)$
D

$2 \cos (α + β)$

Solution

$(\cos α + \cos β)^{2} - (\sin α + \sin β)^{2} = 0$
$\begin{aligned} = (\cos^{2} α + \cos^{2} β + 2 \cos α \cos β) - (\sin^{2} α + \sin^{2} β \\ + 2 \sin α \sin β) = 0 \end{aligned}$
$= \cos 2 α + \cos 2 β = - 2 (\cos α \cos β - \sin α \sin β)$
$= \cos 2 α + \cos 2 β = - 2 \cos (α + β) .$

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