已知sinx+siny=a,cosx+cosy=b,试以a和b表示cos(x-y)

sinx+siny=a,
平方得
sin²x+2sinxsiny+sin²y=a²
cosx+cosy=b
平方得
cos²x+2cosxcosy+cos²y=b²
两式相加得
1+1+2sinxsiny+2cosxcosy=a²+b²
2+2cos(x-y)=a²+b²
cos(x-y)=(a²+b²-2)/2

(sinx+siny)^2=a^2同理那一个式子b^2
cos(x-y)=cosxcosy-sinxsiny=(b^2-a^2-2)/2

a^2+b^2=2+2(sinxsiny+cosxcosy)
cos(x-y)=cosxcosy+sinxsiny=(a^2+b^2-2)/2

sinx+siny=a
两边平方得(sinx)^2+2sinxsiny+(siny)^2=a^2①
cosx+cosy=b
两边平方得(cosx)^2+2cosxcosy+(cosy)^2=b^2②
①+②,利用(sina)^2+(cosa)^2=1得
2+2cosxcosy+2sinxsiny=a^2+b^2
移项得cosxcosy+sinxsiny=1/2(a^2+b^2-2)
利用公式C(A-B)得
cos(x-y)=cosxcosy+sinxsiny=1/2(a^2+b^2-2)

sinx+siny=a,cosx+cosy=b

a^2+b^2=2+2sinxsiny+2cosxcosy=2+2cos(x-y)
cos(x-y)=1/2(a^2+b^2)-1