cosecθ − sinθ = p³ and secθ − cosθ = q³. What is p⁴q² + p²q⁴ equal to?

p³ = cos²θ/sinθ and q³ = sin²θ/cosθ. So p²q² = (p³q³)^(2/3) = (cos²θ · sin²θ / (sinθ cosθ))^(2/3) = (sinθ cosθ)^(2/3). Compute p² + q²: p² = (cos²θ/sinθ)^(2/3), q² = (sin²θ/cosθ)^(2/3). Then p⁴q² + p²q⁴ = p²q²(p² + q²). Using p²q² = (sinθcosθ)^(2/3) and p² + q² = [cos⁴θ/sin²θ]^(1/3) + [sin⁴θ/cos²θ]^(1/3) = (1/(sin²θ cos²θ))^(1/3)·(cos²θ·cos⁴θ/... ). More directly: p²q²(p²+q²) = (sinθcosθ)^(2/3) · [(cos⁴θ)^(1/3)/(sin²θ)^(1/3) + (sin⁴θ)^(1/3)/(cos²θ)^(1/3)] = (sinθcosθ)^(2/3) · (cos⁴θ·cos²θ + sin⁴θ·sin²θ)^(1/3)/(sin²θcos²θ)^(1/3) = (cos⁶θ + sin⁶θ)^(1/3)·(sinθcosθ)^(2/3)/(sinθcosθ)^(2/3)·... After simplification, p⁴q² + p²q⁴ = sin²θ + cos²θ = 1.