Start with the left-hand side:

\(\frac{\sin \theta}{\sin \theta + \cos \theta} + \frac{\cos \theta}{\sin \theta - \cos \theta}\)

Find a common denominator:

\(\frac{\sin \theta (\sin \theta - \cos \theta) + \cos \theta (\sin \theta + \cos \theta)}{(\sin \theta + \cos \theta)(\sin \theta - \cos \theta)}\)

Simplify the numerator:

\(\sin^2 \theta - \sin \theta \cos \theta + \sin \theta \cos \theta + \cos^2 \theta = \sin^2 \theta + \cos^2 \theta\)

Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), the expression becomes:

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

Now consider the right-hand side:

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

Replace \(\tan^2 \theta\) with \(\frac{\sin^2 \theta}{\cos^2 \theta}\):

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

Multiply numerator and denominator by \(\cos^2 \theta\):

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

Using \(\sin^2 \theta + \cos^2 \theta = 1\), this simplifies to:

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

Both sides are equal, hence the identity is proven.