Simplification: 1, sin (x - π / 3) - cos (x + π / 6) + √ 3cosx =? 2, known, sin α + sin β = √ 2 / 2, find the value range of cos α + cos β | Math enthusiast | Mathematical art

Simplification: 1, sin (x - π / 3) - cos (x + π / 6) + √ 3cosx =? 2, known, sin α + sin β = √ 2 / 2, find the value range of cos α + cos β

sin(x-π/3)-cos(x+π/6)+√3cosx
=sinxcosπ/3-cosxsinπ/3-cosxcosπ/6+sinxsinπ/6+√3cosx
=1/2*sinx-√3/2*cosx-√3/2*cosx+1/2*sinx+√3cosx
=sinx
Sin α + sin β = √ 2 / 2, find the value range of cos α + cos β
Sin α + sin β = √ 2 / 2 (square of both sides)
(sinα)^2+(sinβ)^2+2sinαsinβ=1/2.1
Let cos α + cos β = K (square of both sides)
(cosα)^2+(cosβ)^2+2cosαcosβ=k^2.2
1 + 2
SO 2 + 2 (COS α cos β + sin α sin β) = k ^ 2 + 1 / 2
2(cosαcosβ+sinαsinβ)=k^2-3/2
2cos(α-β)=k^2-3/2
cos(α-β)=k^2/2-3/4
-1

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