If (1 + sinθ)/(1 − sinθ) = 9 and θ is acute, then secθ + tanθ equals:

Cross-multiplying, 1 + sinθ = 9 − 9sinθ, so sinθ = 4/5 and cosθ = 3/5. Then secθ + tanθ = 5/3 + 4/3 = 3. The short cut is that $(sec\theta +tan\theta {)}^2$ = ((1 + sinθ)/cosθ)² = (1 + sinθ)/(1 − sinθ), so the answer is simply √9 — forgetting that square root leaves 9. The reciprocal secθ − tanθ is 1/3, and 5/4 is cosecθ.