Start by combining the fractions:

\(\frac{(\tan \theta + 1)(1 - \cos \theta) + (\tan \theta - 1)(1 + \cos \theta)}{(1 + \cos \theta)(1 - \cos \theta)}\)

This simplifies to:

\(\frac{\tan \theta - \tan \theta \cos \theta + 1 - \cos \theta + \tan \theta + \tan \theta \cos \theta - 1 - \cos \theta}{1 - \cos^2 \theta}\)

Combine like terms:

\(\frac{2\tan \theta - 2\cos \theta}{1 - \cos^2 \theta}\)

Since \(1 - \cos^2 \theta = \sin^2 \theta\), the expression becomes:

\(\frac{2(\tan \theta - \cos \theta)}{\sin^2 \theta}\)