Let 12(tanθ + cotθ) = 25, where 45° < θ < 90°. What is a value of (cosecθ + secθ)?

We have sinθ cosθ = 12/25 and sinθ − cosθ = 1/5, so (sinθ + cosθ)² = 1 + 2 × 12/25 = 49/25, giving sinθ + cosθ = 7/5. Then cosecθ + secθ = 1/sinθ + 1/cosθ = (sinθ + cosθ)/(sinθ cosθ) = (7/5)/(12/25) = (7/5) × (25/12) = 35/12.