已知5siny=sin(2x+y),求证:tan(x+y)=3/2tanx

详细解答过程的：

sin[(x+y)+x]=5sin[(x+y)-x]

sin(x+y)·cosx＋cos(x+y)·sinx＝5·sin(x+y)·cosx－5·cos(x+y)·sinx

4·sin(x+y)·cosx＝6·cos(x+y)·sinx

两边同除以cos(x+y)·cosx得:2·tan(x+y)=3·tanx

2x+y=(x+y)+x
y=(x+y)-x 代入原式
5sin(x+y)cosx-5sinxcos(x+y)=sin(x+y)cosx+sinxcos(x+y)
sin(x+y)/cos(x+y)=3sinx/2cosx
得证

相信换元是可以的
令A=2X+Y，B=Y
则5sinB=sinA
即证tan[(A+B)/2]=3/2tan[（A-B）/2]