We start with the left-hand side of the identity:

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

Recognize that \(1 - \sin^2 \theta = \cos^2 \theta\) using the Pythagorean identity.

Substitute to get:

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

Rewrite \(1 - 2 \sin^2 \theta\) as \(\cos^2 \theta - \sin^2 \theta\) using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\).

Thus, the expression becomes:

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

Separate the fraction:

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

This simplifies to:

\(1 - \tan^2 \theta\).

Thus, the identity is proven.