If p = 1/(cosec θ + cot θ) and q = cosec θ, then what is p² − 2pq equal to?

p = 1/(cosec θ + cot θ) = cosec θ − cot θ (rationalizing, since (cosec θ + cot θ)(cosec θ − cot θ) = cosec²θ − cot²θ = 1). So p = cosec θ − cot θ. Then p² − 2pq = (cosec θ − cot θ)² − 2(cosec θ − cot θ)(cosec θ) = cosec²θ − 2cosec θ·cot θ + cot²θ − 2cosec²θ + 2cosec θ·cot θ = cot²θ − cosec²θ = −1.