p + q cotθ = 3cosecθ and q − p cotθ = 2cosecθ. What is p² + q² equal to?

Square both equations: (p + q cotθ)² = 9cosec²θ and (q − p cotθ)² = 4cosec²θ. Adding: p² + 2pq cotθ + q²cot²θ + q² − 2pq cotθ + p²cot²θ = 13cosec²θ. This gives (p² + q²)(1 + cot²θ) = 13cosec²θ. Since 1 + cot²θ = cosec²θ, we get (p² + q²)cosec²θ = 13cosec²θ, so p² + q² = 13.