Multiply top and bottom by 1+cosθ+sinθ:
(1+cosθ+sinθ)2/((1+cosθ)2-sin2θ)=
(1+2cosθ+cos2θ+sin2θ+2sinθ(1+cosθ))/(1+2cosθ+cos2θ-sin2θ)=
(1+2cosθ+1+2sinθ(1+cosθ)/(2cosθ+cos2θ+cos2θ)=
(2(1+cosθ)+2sinθ(1+cosθ)/(2cosθ+2cos2θ)=
(1+cosθ)(1+sinθ)/(cosθ(1+cosθ))=
(1+sinθ)/cosθ=secθ+tanθ QED