Find the area of the greatest rectangle that can be inscribed in an ellipse x²/a² + y²/b² = 1.

The vertices of any rectangle inscribed in an ellipse is given by (±acosθ, ±bsinθ). The area of the rectangle is given by A(θ) = 4abcosθsinθ = 2ab·sin(2θ). Hence, the maximum is when sin2θ = 1. Hence, the maximum area is when 2θ = π/2 i.e. θ = π/4. The maximum area is A = 2ab.