Start with the left-hand side: \(\tan^2 \theta - \sin^2 \theta\).

Using \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we have \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\).

Substitute into the expression: \(\frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta\).

Factor out \(\sin^2 \theta\): \(\sin^2 \theta \left( \frac{1}{\cos^2 \theta} - 1 \right)\).

Using \(1 - \cos^2 \theta = \sin^2 \theta\), rewrite as \(\sin^2 \theta \left( \frac{\sin^2 \theta}{\cos^2 \theta} \right)\).

This simplifies to \(\tan^2 \theta \sin^2 \theta\), which is the right-hand side.