cos²α+cos²β+sinαsinβ
=[1+cos(2α)]/2+[1+cos(2β)]/2+sinαsinβ
=1+[cos(2α)+cos(2β)]/2+sinαsinβ
=1+2cos(α+β)cos(α-β)/2 +sinαsinβ/运用了和差化积公式
=1+cos(α+β)cos(π/3)+sinαsinβ
=1+(1/2)cos(α+β)+sinαsinβ
=1+(1/2)(cosαcosβ-sinαsinβ)+sinαsinβ
=1+(1/2)(cosαcosβ+sinαsinβ)
=1+(1/2)cos(α-β)
=1+(1/2)cos(π/3)
=1+(1/2)(1/2)
=1+1/4
=5/4