The product (secθ + tanθ)(1 − sinθ) simplifies to:

secθ + tanθ = (1 + sinθ)/cosθ, so the product is (1 + sinθ)(1 − sinθ)/cosθ = (1 − si$n^2$θ)/cosθ = co$s^2$θ/cosθ = cosθ. Stopping one step early, at (1 − sinθ)/cosθ, gives the trap secθ − tanθ.