Concept:Use trigonometric identities to simplify expressions. Express p in terms of sine and cosine, then combine terms.
Explanation:First, rewrite p: p = 1/(cosecθ + cotθ). Since cosecθ = 1/sinθ and cotθ = cosθ/sinθ, we have p = 1/((1+cosθ)/sinθ) = sinθ/(1+cosθ).
Given q = cosecθ = 1/sinθ.
Now compute p² - 2pq: substitute values: (sinθ/(1+cosθ))² - 2 × (sinθ/(1+cosθ)) × (1/sinθ) = sin²θ/(1+cosθ)² - 2/(1+cosθ).
Use identity sin²θ = 1 - cos²θ = (1 - cosθ)(1 + cosθ). So expression becomes (1 - cosθ)(1+cosθ)/(1+cosθ)² - 2/(1+cosθ) = (1 - cosθ)/(1+cosθ) - 2/(1+cosθ).
Combine numerators: (1 - cosθ - 2)/(1+cosθ) = (-1 - cosθ)/(1+cosθ) = -(1+cosθ)/(1+cosθ) = -1.
Answer:-1 (Option A)