p = sinθ/(1 + cosθ + sinθ) and q = (1 + sinθ)/(1 + sinθ − cosθ). Which one of the following is correct?

Multiply numerator and denominator of p by (1 + sinθ − cosθ): p × (1+sinθ−cosθ)/(1+sinθ−cosθ). Actually, a standard identity shows that sinθ/(1+cosθ+sinθ) = (1+sinθ−cosθ)/(1+sinθ+cosθ−cosθ−... ) — simpler: take θ = 90°: p = 1/(1+0+1) = 1/2; q = (1+1)/(1+1−0) = 2/2 = 1. So p ≠ q here, ruling out (a). Recheck: pq − 1 = 1/2 − 1 = −1/2; 2pq − 1 = 0. So 2pq − 1 = 0 is the correct identity. Therefore option (b) is correct.