If ta$n^2$θ = $1-e^2$, then secθ + ta$n^3$θ·cosecθ equals:

$se$$c^2$θ = 1 + ta$n^2$θ = $2-e^2$, so secθ = √$(2-e^2)$. Next, ta$n^3$θ·cosecθ = (si$n^3$θ/co$s^3$θ)·(1/sinθ) = si$n^2$θ/co$s^3$θ = (1 − co$s^2$θ)/co$s^3$θ = $se$$c^3$θ − secθ. Adding secθ leaves $se$$c^3$θ = $(2-e^2)$√$(2-e^2)$.