Let 12(tanθ + cotθ) = 25, where 45° < θ < 90°. What is the value of (sinθ − cosθ)?

tanθ + cotθ = 25/12, which gives (sin²θ + cos²θ)/(sinθ cosθ) = 25/12, so sinθ cosθ = 12/25. Then (sinθ − cosθ)² = 1 − 2sinθ cosθ = 1 − 24/25 = 1/25. So sinθ − cosθ = ±1/5. Since 45° < θ < 90°, sinθ > cosθ, hence sinθ − cosθ = 1/5.