Find out the answer to "If cos⁡αcos⁡β+sin⁡αsin⁡β=−1, find cos3βcos⁡α+sin3βsin⁡α"

micelije1mw

Answered question

2022-04-21

\frac{\cos α}{\cos β} + \frac{\sin α}{\sin β} = - 1

, find

\frac{\cos^{3} β}{\cos α} + \frac{\sin^{3} β}{\sin α}

Melissa Collins

Beginner2022-04-22Added 12 answers

(\frac{\cos α}{\cos β} + \frac{\sin α}{\sin α} + 1 = 0), (\sin 2 α + \sin 2 β = 2 \sin (α + β) \cos (α - β)) :

(2 \sin (α + β) = - \sin 2 β), (2 \sin (α + β) (1 + \cos (α - β)) = \sin 2 α) :

\frac{\cos^{3} β}{\cos α} + \frac{\sin^{3} β}{\sin α} = \frac{3 \cos β + \cos 3 β}{4 \cos α} + \frac{3 \sin β - \sin 3 β}{4 \sin α} =

= \frac{3 \sin (α + β) + \sin (α - 3 β)}{2 \sin 2 α} = \frac{4 \sin (α + β) - (\sin (α + β) - \sin (α - 3 β))}{2 \sin 2 α} = \frac{4 \sin (α + β) - 2 \sin 2 β \cos (α - β)}{2 \sin 2 α} = \frac{2 \sin (α + β) (1 + \cos (α - β))}{\sin 2 α} =

= \frac{\sin 2 α}{\sin 2 α} = 1

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