cosx+cosy=1/2 ===> (cosx+cosy)^2=1/4 ===> cos^2 x+2cosxcosy+cos^2 y=1/4 sinx+siny=1/3 ===> (sinx+siny)^2=1/9 ===> sin^2 x+2sinxsiny+sin^2 y=1/9 两式相加有：(sin^2 x+cos^2 x)+2(cosxcosy+sinxsiny)+(sin^2 y+cos^2 y)=(1/4)+(1/9) ===> 1+2cos(x-y)+1=13/36 ===> 2cos(x-y)=(13/36)-2=-59/36 ===> cos(x-y)=-59/72 cosx+cosy=1/2 ===> 2cos[(x+y)/2]*cos[(x-y)/2]=1/2 sinx+siny=1/3 ===> 2sin[(x+y)/2]*cos[(x-y)/2]=1/3 两式相除有：tan[(x+y)/2]=(1/3)/(1/2)=2/3 所以，cos(x+y)=[1-tan^2(x+y/2)]/[1+tan^2(x+y/2)]=[1-(4/9)]/[1+(4/9)]=5/13。
sinx+siny=2sin[(x+y)/2]cos[(x-y)/2]=1/3 ① cosx+cosy=2cos[(x+y)/2]cos[(x-y)/2]=1/2 ② ①÷②，得 tan[(x+y)/2]=2/3, 则 sin[(x+y)/2]=2/√13 ③ 代入①，得 cos[(x-y)/2]=√13/12 ④ 依据公式 1+cos2α=2(cosα)^2, 由④，得 cos(x-y)=2[cos(x-y)/2]^2-1=-59/72, 依据公式 1-cos2α=2(sinα)^2, 由③，得 cos(x+y)=1-2[sin(x+y)/2]^2=5/13.
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