以知tanX/2=2/3,則(1-cosX+sinX)/(1+cosX+sinX)的值是多少？

熱心網友

用萬能公式：sin(x)=2sin(x/2)cos(x/2)=2[sin(x/2)/cos(x/2)]*1/[sec(x/2)]^2=2*tan(x/2)/[1+(tan(x/2))^2]（前面部分是推導過程）tan(x)=2*tan(x/2)/[1-(tan(x/2))^2]（二倍角公式）cos(x)=sin(x)/tan(x)=[1-(tan(x/2))^2]/[1+(tan(x/2))^2](1-cosX+sinX)/(1+cosX+sinX)=[1-(1-4/9)/(1+4/9)+(4/3)/(1+4/9)]/[1+(1-4/9)/(1+4/9)+(4/3)/(1+4/9)]=(1-5/13+12/13)/(1+5/13+12/13)=20/30=2/3

熱心網友

(1-cosx+sinx)/(1+cosx+sinx)={2[sin(x/2)]^2+2sin(x/2)cos(x/2)}/{2[cos(x/2)]^2+2sin(x/2)cos(x/2)}=2sin(x/2)[sin(x/2+cos(x/2)]/{2cos(x/2)[cos(x/2)+sin(x/2)]=sin(x/c)/cos(x/2)=tan(x/2)=2/3.