LEFT-HAND SIDE
(tanθ+secθ-1)/(tanθ-secθ+1). Multiply top and bottom by cosθ:
(sinθ+1-cosθ)/(sinθ-1+cosθ). Multiply top and bottom by sinθ+1-cosθ
(sinθ+1-cosθ)2/(sin2θ-(1-cosθ)2)=
(sin2θ+2sinθ+1-2cosθ(1+sinθ)+cos2θ)/(sin2θ-1+2cosθ-cos2θ)=
(2+2sinθ-2cosθ(1+sinθ))/(-cos2θ+2cosθ-cos2θ)=
(2+2sinθ-2cosθ(1+sinθ))/(2cosθ-2cos2θ)=
((1+sinθ)-cosθ(1+sinθ))/(cosθ(1-cosθ))=
(1-cosθ)(1+sinθ)/(cosθ(1-cosθ))=(1+sinθ)/cosθ=secθ+tanθ.
RIGHT-HAND SIDE
cosθ/(1-sinθ)=cosθ(1+sinθ)/(1-sin2θ)=cosθ(1+sinθ)/cos2θ=secθ+tanθ.
Therefore, (tanθ+secθ-1)/(tanθ-secθ+1)=cosθ/(1-sinθ) QED