求证(2cosθ-sin2θ)/(2cosθ+sin2θ)=tan(π/4-θ/2)右边应该是tan(π/4-θ/2)^2

来源:学生作业帮助网 编辑:作业帮 时间:2024/05/08 20:23:28
求证(2cosθ-sin2θ)/(2cosθ+sin2θ)=tan(π/4-θ/2)右边应该是tan(π/4-θ/2)^2

求证(2cosθ-sin2θ)/(2cosθ+sin2θ)=tan(π/4-θ/2)右边应该是tan(π/4-θ/2)^2
求证(2cosθ-sin2θ)/(2cosθ+sin2θ)=tan(π/4-θ/2)
右边应该是tan(π/4-θ/2)^2

求证(2cosθ-sin2θ)/(2cosθ+sin2θ)=tan(π/4-θ/2)右边应该是tan(π/4-θ/2)^2
(2cosθ-sin2θ)/(2cosθ+sin2θ)=2cosθ(1-sinθ)/[2cosθ(1+sinθ)]=(1-sinθ)/(1+sinθ)
tan(π/4-θ/2)=sin(π/4-θ/2)/cos(π/4-θ/2)=(cosθ/2-sinθ/2)/(sinθ/2+cosθ/2)
=根号(cosθ/2-sinθ/2)^2/(sinθ/2+cosθ/2)^2=根号(1-sinθ)/(1+sinθ)
所以(2cosθ-sin2θ)/(2cosθ+sin2θ)=[tan(π/4-θ/2)]^2

(2cosθ-sin2θ)/(2cosθ+sin2θ)分子分母同时提取2cosθ,等式左边等于:
(1-sinθ)/(1+sinθ)=((cosθ/2)^2+(sinθ/2)^2-2sinθ/2*cosθ/2)/((cosθ/2)^2+(sinθ/2)^2+2sinθ/2*cosθ/2)=((cosθ/2+sinθ/2)/(cosθ/2-sinθ/2))^2=((1+tanθ/2)/(1-tanθ/2))^2=tan(π/4-θ/2)^2;得证