Problem 7 - Entrance Test

Rewrite the expression in terms of sin θ and cos θ: (1 + sin θ/cos θ + 1/cos θ)(1 + cos θ/sin θ - 1/sin θ) = ((cos θ + sin θ + 1)/cos θ)((sin θ + cos θ - 1)/sin θ) Let (sin θ + cos θ) = X. The expression becomes: = ((X + 1)/cos θ)((X - 1)/sin θ) = (X² - 1) / (sin θ cos θ) Substitute X back: ( (sin θ + cos θ)² - 1 ) / (sin θ cos θ) Expand (sin θ + cos θ)²: (sin² θ + cos² θ + 2 sin θ cos θ - 1) / (sin θ cos θ) Since sin² θ + cos² θ = 1: = (1 + 2 sin θ cos θ - 1) / (sin θ cos θ) = (2 sin θ cos θ) / (sin θ cos θ) = 2.