Start with the left-hand side: \(\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{1 + \cos \theta}\).

Let \(c = \cos \theta\) and \(s = \sin \theta\).

Combine the fractions: \(\frac{(1+c)^2 + s^2}{s(1+c)}\).

Expand the numerator: \((1+c)^2 + s^2 = 1 + 2c + c^2 + s^2\).

Use the identity \(c^2 + s^2 = 1\), so the numerator becomes \(1 + 2c + 1 = 2 + 2c\).

Thus, \(\frac{2 + 2c}{s(1+c)} = \frac{2(1+c)}{s(1+c)}\).

Cancel \(1+c\) from the numerator and denominator: \(\frac{2}{s}\).

This matches the right-hand side: \(\frac{2}{\sin \theta}\).