sinα+sinβ=1/4
(sinα+sinβ)²=1/16
sin²α+2sinαsinβ+sin²β=1/16
cosα+cosβ=1/3
(cosα+cosβ)²=1/9
cos²α+2cosαcosβ+cos²β=1/9
两式相加
sin²α+2sinαsinβ+sin²β+cos²α+2cosαcosβ+cos²β=1/16+1/9
(sin²α+cos²α)+(sin²β+cos²β)+2(cosαcosβ+sinαsinβ)=25/144
2+2cos(α-β)=25/144
cos(α-β)=-263/288
两式相减
cos²α+2cosαcosβ+cos²β-(sin²α+2sinαsinβ+sin²β)=1/9-1/16
(cos²α-sin²α)+(cos²β-sin²β)+2(cosαcosβ-sinαsinβ)=7/144
cos(2α)+cos(2β)+2cos(α+β)=7/144
2cos(α+β)cos(α-β)+2cos(α+β)=7/144
[1+cos(α-β)]cos(α+β)=7/288
(1-263/288)cos(α+β)=7/288
cos(α+β)=7/25
原式相乘
(sinα+sinβ)(cosα+cosβ)=1/12
sinαcosα+sinβcosβ+sinαcosβ+cosαsinβ=1/12
[sin(2α)+sin(2β)]/2+sin(α+β)=1/12
sin(α+β)cos(α-β)+sin(α+β)=1/12
[1+cos(α-β)]sin(α+β)=1/12
(1-263/288)sin(α+β)=1/12
sin(α+β)=24/25
得sin(α+β)=24/25、cos(α+β)=7/25