当α+β=30度时,求sinα平方+cosβ平方+cosαsinβ的值

∵α+β=30°,∴α=30°-β
(sinα)^2=[sin(30°-β)]^2=(sin30°.cosβ-cos30°.sinβ)^2
={(1/2)cosβ-[3^(1/2)]/2*sinβ}^2
=(1/4)(cosβ)^2-[3^(1/2)]/2*cosβ.sinβ+(3/4)(sinβ)^2
原式=(sinα)^2+(cosβ)^2+cosαsinβ
=(1/4)(cosβ)^2-[3^(1/2)]/2*cosβ.sinβ+(3/4)(sinβ)^2+(cosβ)^2+cosαsinβ
=(3/4)[(cosβ)^2+(sinβ)^2]+(1/2)(cosβ)^2-[3^(1/2)]/2*cosβ.sinβ+cosαsinβ
=3/4+(cosβ){(1/2)(cosβ)-[3^(1/2)]/2*sinβ}+cosαsinβ
=3/4+cosβsin(30°-β)+cosαsinβ
=3/4+sinα.cosβ+cosαsinβ
=3/4+sin(α+β)
=3/4+sin30°=3/4+1/2
=5/4

a+β=30°，则β=30°-a
sina²+cosβ²+cosasinβ=sina²+½+½cos（60°+2a）+sinasin（30°-a）
=½-½cos2a+½[½cos2a-½（根3）sin2a]+½（根3）sina cosa-½sina²...