证明等式:(1-cosx+sinx)/(1+sinx+cosx)=(sinx)/(1+cosx)
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证明等式:(1-cosx+sinx)/(1+sinx+cosx)=(sinx)/(1+cosx)
证明等式:(1-cosx+sinx)/(1+sinx+cosx)=(sinx)/(1+cosx)
证明等式:(1-cosx+sinx)/(1+sinx+cosx)=(sinx)/(1+cosx)
用反证法:
假设该等式成立,则:
(1-cosx+sinx)/(1+sinx+cosx)=(sinx)/(1+cosx)
(1-cosx+sinx)*(1+cosx)= (1+sinx+cosx)*(sinx)
1-cosx+sinx+cosx-(cosx)^2+sinxcosx=sinx+sinx^2+sinxcosx
约掉相同的
1-cosx^2=sinx^2
1=sinx^2+cosx^2
所以假设成立,即该等式成立
1-cosx+sinx)/(1+sinx+cosx)=sinx/(1+cosx)。将cosx用半角公式
左边=[2sin²(x/2)+2sin(x/2)cos(x/2)]/[2cos²(x/2)+2sin(x/2)cos(x/2)]
=2sin(x/2)[sin(x/2)+cos(x/2)]/ 2cos(x/2)[sin(x/2)+cos(x/2)]...
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1-cosx+sinx)/(1+sinx+cosx)=sinx/(1+cosx)。将cosx用半角公式
左边=[2sin²(x/2)+2sin(x/2)cos(x/2)]/[2cos²(x/2)+2sin(x/2)cos(x/2)]
=2sin(x/2)[sin(x/2)+cos(x/2)]/ 2cos(x/2)[sin(x/2)+cos(x/2)]
=sin(x/2)/ cos(x/2)
=2sin(x/2)cos(x/2)/cos²(x/2) 再反过来用倍角
=sinx / (1+cosx)
=右边
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