To solve the given problem, we need to determine which mathematical expression among the options is correct based on the provided values of \( p \) and \( q \):

Given: \(p = \frac{\sin \theta}{1 + \cos \theta + \sin \theta}\) and \(q = \frac{1 + \sin \theta}{1 + \sin \theta - \cos \theta}\)

1. Substituting \( p \) and \( q \) into the expressions:
   • \(p - q = \frac{\sin \theta}{1 + \cos \theta + \sin \theta} - \frac{1 + \sin \theta}{1 + \sin \theta - \cos \theta}\)
   • \(pq - 1 = \left(\frac{\sin \theta}{1 + \cos \theta + \sin \theta}\right) \left(\frac{1 + \sin \theta}{1 + \sin \theta - \cos \theta}\right) - 1\)
   • \(pq - 2 = \left(\frac{\sin \theta}{1 + \cos \theta + \sin \theta}\right) \left(\frac{1 + \sin \theta}{1 + \sin \theta - \cos \theta}\right) - 2\)
   • \(2pq - 1 = 2\left(\frac{\sin \theta}{1 + \cos \theta + \sin \theta}\right) \left(\frac{1 + \sin \theta}{1 + \sin \theta - \cos \theta}\right) - 1\)
2. Calculate \( pq \):

First, find the product \( pq \):

\(pq = \left(\frac{\sin \theta}{1 + \cos \theta + \sin \theta}\right) \left(\frac{1 + \sin \theta}{1 + \sin \theta - \cos \theta}\right)\)

By simplifying the expression \( pq = \frac{\sin \theta (1 + \sin \theta)}{(1 + \cos \theta + \sin \theta)(1 + \sin \theta - \cos \theta)} \).

Notice that the denominator may be simplified using algebraic identities.

After simplification, we find that \(2pq - 1 = 0\) turns out to be true.

1. Verification:

Substitute and simplify the expression to check the validity:

This involves simplification of trigonometric identities which leads to verifying:

\(2pq = 1 \implies \frac{2 \sin \theta (1 + \sin \theta)}{(1 + \cos \theta + \sin \theta)(1 + \sin \theta - \cos \theta)} = 1\)

Upon finding these values and verifying all parts, we conclude:

1. Conclusion:

Based on the calculations and verification, the correct answer is indeed:

Option: \(2pq - 1 = 0\)