If (sin θ - cos θ + 1)/(sin θ + cos θ - 1) = p sec θ + q tan θ, where 0 < θ < π/2, then what is p + q equal to?

Multiply numerator and denominator by (sin θ + cos θ + 1). Numerator × this = (sin θ + 1)² - cos²θ = sin²θ + 2sin θ + 1 - cos²θ = 2sin²θ + 2sin θ = 2sin θ(sin θ + 1). Denominator × this = (sin θ + cos θ)² - 1 = 2 sin θ cos θ. Ratio = 2sinθ(sinθ+1)/(2sinθcosθ) = (sinθ + 1)/cosθ = tan θ + sec θ. So p = 1, q = 1, p + q = 2.