Explanation

This is an AM–GM optimization. For positive terms, 9tan²θ + 4cot²θ ≥ 2√(9tan²θ · 4cot²θ) = 2 × 6 = 12, since tan²θ·cot²θ = 1. Equality holds when 9tan²θ = 4cot²θ (i.e. 3tanθ = 2cotθ), which is attainable, so the minimum is 12. Option (a) 6 is the trap from forgetting the factor 2 in AM–GM. Hence option (c).