Question

已知：α-β=π/3,证明 cos^2α+cos^2β+sinαsinβ为定值,谢谢

Answer 1

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