Explanation

Write cotθ − cosecθ = (cosθ−1)/sinθ. Then the product is ((cosθ−sinθ+1)(cosθ−1))/((cosθ+sinθ−1)sinθ). At θ = 60° this is (√3)(−1/√3) = −1, and at θ = 30° the same −1. The expression identically equals −1 — option A.