The value of $$\frac{\sec^6 \theta - \tan^6 \theta - 3\sec^2 \theta \tan^2 \theta + 1}{\cos^4 \theta - \sin^4 \theta + 2 \sin^2 \theta + 2}$$ is:

$$\frac{\sec^6 \theta - \tan^6 \theta - 3\sec^2 \theta \tan^2 \theta + 1}{\cos^4 \theta - \sin^4 \theta + 2 \sin^2 \theta + 2}$$

= $$\frac{\sec^6 \theta - \tan^6 \theta - 3\sec^2 \theta \tan^2 \theta(\sec^2 \theta - \tan^2 \theta) + 1}{(\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta) + 2 \sin^2 \theta + 2}$$

($$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$$)

($$a^2 - b^2 = (a+b)(a-b)$$)

= $$\frac{(\sec^6 \theta - \tan^6 \theta)^3 + 1}{\cos^2 \theta - \sin^2 \theta + 2 \sin^2 \theta + 2}$$

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

= $$\frac{2}{1 + 2}$$

= $$\frac{2}{3}$$