高一数学三角函数已知:x、y是锐
因为sin2x/sin(2x+y)=sin2y/sin(2y+x),
所以sin(2x+y)/sin2x=sin(2y+x)/sin2y (*)
sin(2x+y)=sin2xcosy+sinycos2x
sin(2y+x)=sin2ycosx+sinxcos2y
所以(*)式可化为cosy+sinxcot2x=cosx+sinycot2y (**)
cot2x=cos2x/sin2x=cos2x/(2sinxcosx)
cot2y=cos2y/sin2y=cos2x/(2sinycosy)
代入(**)式,可得
cosy+cos2x/(2cosx)=cosx+cos2y/(2cosy...全部
因为sin2x/sin(2x+y)=sin2y/sin(2y+x),
所以sin(2x+y)/sin2x=sin(2y+x)/sin2y (*)
sin(2x+y)=sin2xcosy+sinycos2x
sin(2y+x)=sin2ycosx+sinxcos2y
所以(*)式可化为cosy+sinxcot2x=cosx+sinycot2y (**)
cot2x=cos2x/sin2x=cos2x/(2sinxcosx)
cot2y=cos2y/sin2y=cos2x/(2sinycosy)
代入(**)式,可得
cosy+cos2x/(2cosx)=cosx+cos2y/(2cosy)
将等式两边同乘以2cosxcosy,
则2cosxcosy+cos2x=2cosxcosy+cos2y
cos2x=cos2y
倍角公式
(sinx)2=(siny)2
又因为x,y<π/2
所以sinx,siny 〉0
所以sinx=siny
又根据正弦函数在(0,π/2]单调,可知x=y
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