高中三角函数证明题

来源：百度知道 编辑：UC知道 时间：2024/05/06 08:11:11
证明:sin^2(x)+sin^2(y)+2*sin*(x)*sin(y)cos(x+y)=sin^2(x+y)
要最简便的方法。。

sin^2(x)+sin^2(y)+2*sin(x)*sin(y)cos(x+y)

= (1-cos2x+1-cos2y)/2+2*sin(x)*sin(y)cos(x+y)

= 1 -cos(x+y)cos(x-y)+2*sin(x)*sin(y)cos(x+y)

= 1-cos(x+y)[cos(x-y)+2*sin(x)sin(y)] = 1-cos^2(x-y)=sin^2(x+y)

sin(x)*sin(y)*cos(x+y)=sin(x)*sin(y)*[cos(x)*cos(y)-sin*(x)*sin(y)]=
sin(x)*sin(y)*cos(x)*cos(y)-sin^2(x)*sin^2(y)

sin^2(x)+sin^2(y)+2*sin(x)*sin(y)*cos(x+y)=
sin^2(x)+sin^2(y)+2sin(x)*sin(y)*cos(x)*cos(y)-2sin^2(x)*sin^2(y)=
sin^2(x)+sin^2(y)-2sin^2(x)*sin^2(y)+2sin(x)*sin(y)*cos(x)*cos(y)=
sin^2(x)+sin^2(y)-sin^2(x)*sin^2(y)-sin^2(x)*sin^2(y)+2sin(x)*sin(y)*cos(x)*cos(y)=
[1*sin^2(x)-sin^2(x)*sin^2(y)]+[1*sin^2(y)-sin^2(x)*sin^2(y)]+2sin(x)*sin(y)*cos(x)*cos(y)=
sin^2(x)*cos^2(x)+sin^2(y)*cos^2(y)+2sin(x)*sin(y)*cos(x)*cos(y)=
[sin(x)*cos(y)+cos(x)*sin(y)]^2=sin^2(x+y).