If sin4α+4cos4β+2=42sin α cosβ, α,β∈0,π then cosα+β-cosα-β=

sin4α+4cos4β+=42 sin α cos β AM≥GM sin4α+4cos4α+1+14≥sin4α 4cos4α.1.114 AM=GM sin4α=4cos4β=1 sin α=1, cos β=±12 sin β=12 β∈0, π cos(α+β)-cos(α-β)=-2sin α sin β =-2(1)12 =-2

If sin4α+4cos4β+2=42sin α cosβ, α,β∈0,π then cosα+β-cosα-β=

sin4α+4cos4β+=42 sin α cos β AM≥GM sin4α+4cos4α+1+14≥sin4α 4cos4α.1.114 AM=GM sin4α=4cos4β=1 sin α=1, cos β=±12 sin β=12 β∈0, π cos(α+β)-cos(α-β)=-2sin α sin β =-2(1)12 =-2

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If $\sin^{4} α + 4 \cos^{4} β + 2 = 4 \sqrt{2} \sin α cosβ,$ $α, β \in [0, π]$ then $\cos (α + β) - \cos (α - β) =$

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$- \sqrt{2}$

-1

$\sqrt{2}$

answer is A.

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Detailed Solution

$\sin^{4} α + 4 \cos^{4} β + = 4 \sqrt{2} \sin α \cos β AM \geq GM \frac{\sin^{4} α + 4 \cos^{4} α + 1 + 1}{4} \geq {(\sin^{4} α 4 \cos^{4} α . 1.1)}^{\frac{1}{4}} AM = GM \sin^{4} α = 4 \cos^{4} β = 1 \sin α = 1, \cos β = \pm \frac{1}{\sqrt{2}} \sin β = \frac{1}{\sqrt{2}} β \in [0, π] \cos (α + β) - \cos (α - β) = - 2 \sin α \sin β = - 2 (1) \frac{1}{\sqrt{2}} = - \sqrt{2}$