已知sin(x+2y)=3sinx,求tan(x+y)*coty

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已知sin(x+2y)=3sinx,求tan(x+y)*coty
已知sin(x+2y)=3sinx,求tan(x+y)*coty

已知sin(x+2y)=3sinx,求tan(x+y)*coty
sin(x+2y)=3sinx ,sin[(x+y)+y]=3sin[(x+y)-y] ,sin(x+y)cos(y)+cos(x+y)sin(y)=3[sin(x+y)cos(y)-cos(x+y)sin(y)] ,sin(x+y)cos(y)=2cos(x+y)sin(y) ,tan(x+y)*coty=2

sin(x+y + y) = 3sin(x+y - y)
sin(x+y)cosy + cos(x+y)siny = 3[sin(x+y)cosy - cos(x+y)siny],
4cos(x+y)siny = 2sin(x+y)cosy,
2siny/cosy = sin(x+y)/cos(x+y),
2tany = tan(x+y),
2 = tan(x+y)coty

sin(x+2y)=3sinx
sinxcos2y+cosxsin2y=3sinx（拆开sin(x+2y)）
cos2y+sin2ycotx=3(两边同除以sinx)
1-2(siny)^2+2sinycosycotx=3(拆开cos2y和sin2y)
sinycosycotx=(cosy)^2(移项整理)
sinycotx=cosy
cotx=coty
后面把这个结论带进去就行了。

Tan[x + y]*Cot[y] =
(Sin[x + y]*Cos[y])/( Cos[x + y]*Sin[y]) =
(1/2 (Sin[x] + Sin[x + 2 y]))/(1/2 (-Sin[x] + Sin[x + 2 y])) =
(Sin[x] + Sin[x + 2 y])/ (-Sin[x] + Sin[x + 2 y]) =
(Sin[x] + 3 Sin[x])/ (-Sin[x] + 3 Sin[x]) =
4/2 =
2

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