(sin θ − cosec θ)(cos θ − sec θ) is equal to

(si$n^2$θ − 1)/sin θ × (co$s^2$θ − 1)/cos θ = (co$s^2$θ si$n^2$θ)/(sin θ cos θ) = sin θ cos θ, and sin θ cos θ = 1/(tan θ + cot θ).