Explanation

With 0 ≤ α−β ≤ α+β ≤ π/2, tan(α+β) = √3 gives α+β = 60°, and tan(α−β) = 1/√3 gives α−β = 30°. Solving: α = 45°, β = 15°. Then tanα·cot2β = tan45°·cot30° = 1×√3 = √3 — option C.