cosecθ=1/sinθ, cotθ=cosθ/sinθ, so:
cosecθ+cotθ=(1+cosθ)/sinθ and cosecθ-cotθ=(1-cosθ)/sinθ.
Therefore (cosecθ+cotθ)/(cosecθ-cotθ)=(1+cosθ)/(1-cosθ).
Multiply top and bottom by 1+cosθ:
(1+cosθ)2/(1-cos2θ)=(1+cosθ)2/sin2θ=
(1+2cosθ+cos2θ)/sin2θ=csc2θ+2cotθcscθ+cot2θ.
sin2θ+cos2θ≡1, so 1+cot2θ≡csc2θ.
(cosecθ+cotθ)/(cosecθ-cotθ)=1+cot2θ+2cotθcscθ+cot2θ=
1+2cot2θ+2cscθcotθ QED