2026 · Q41
If cot θ + cos θ = m, cot θ − cos θ = n, where 0 < θ < π/2, then what is (m²−n²)/√(mn) equal to?
Elementary MathematicsTrigonometry
- A1
- B2
- C2√2
- D4
Explanation
Compute m² − n² = (m − n)(m + n) = (2cos θ)(2cot θ) = 4cos θcot θ. Also mn = cot²θ − cos²θ = cos²θ(1/sin²θ − 1) = cos⁴θ/sin²θ, so √(mn) = cos²θ/sin θ (positive on (0, π/2)). The ratio is 4cos θ·(cos θ/sin θ)·(sin θ/cos²θ) = 4, option D.