The value of $$\frac{\tan^2 \theta - \sin^2 \theta}{2 + \tan^2 \theta + \cot^2 \theta}$$ is:

$$\frac{\tan^2 \theta - \sin^2 \theta}{2 + \tan^2 \theta + \cot^2 \theta}$$

= $$\frac{\tan^2 \theta - \sin^2 \theta}{1 + \tan^2 \theta + 1 + \cot^2 \theta}$$

= $$\frac{\tan^2 \theta - \sin^2 \theta}{\sec^2 \theta + \cosec^2 \theta}$$

= $$\frac{\tan^2 \theta - \sin^2 \theta}{\frac{1}{\cos^2 \theta} +\frac{1}{\sin^2 \theta}}$$

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

= $$(\tan^2 \theta - \sin^2 \theta)(\sin^2\theta\cos^2 \theta)$$

= $$ \sin^4 \theta - \sin^4 \theta\cos^2 \theta$$

= $$ \sin^4 \theta(1 - \cos^2 \theta)$$ = $$ \sin^4 \theta \sin^2 \theta =  \sin^6 \theta$$