tan(x+y)=2tany,求证3sinx=sin(x+2y)

tan(x+y)=2tany,求证3sinx=sin(x+2y)
tan(x+y)=2tany,求证3sinx=sin(x+2y)

tan(x+y)=2tany,求证3sinx=sin(x+2y)
证明：
因为tan(x+y)=2tany
所以,sin(x+y)÷cos(x+y)=2siny÷cosy
所以,sin(x+y)cosy=2cos(x+y)siny
所以,2sin(x+y)cosy=4cos(x+y)siny
所以,3sin(x+y)cosy-3cos(x+y)siny=sin(x+y)cosy+cos(x+y)siny
所以,3sin[(x+y)-y]=sin[(x+y)+y]
所以,3sinx=sin(x+2y)

3sinx=sinxcos2y+sin2ycosx
sinx(3-cos2y)=sin2ycosx
tanx=sin2y/(3-cos2y)
tanx+tany=2tany(1-tanxtany)
tanx[1+2(tany)^2]=tany
tanx=tany/[1+2(tany)^2]=sinycosy/[(cosy)^2+2(siny)^2]=sin2y/(3-cos2y)

就是要证明：3sin[(x+y)-y]=sin[(x+y)+y]
也就是要证明：3sin(x+y)cosy-3sinycos(x+y)=sin(x+y)cosy+sinycos(x+y)
即证明：sin(x+y)cosy＝2sinycos(x+y)
即：tan(x+y)=2tany
结论显然成立