If $$7 \cos^2 \theta + 3 \sin^2 \theta = 6, 0^\circ < \theta < 90^\circ,$$ then the value of $$\frac{\cot^2 2\theta + \sec^2 2\theta}{\tan^2 2\theta - \sin^2 2\theta}$$ is:

$$7 \cos^2 \theta + 3 \sin^2 \theta = 6$$

$$7(1 - \sin^2\theta) + 3 \sin^2 \theta = 6$$

$$7 - 7\sin^2\theta + 3 \sin^2 \theta = 6$$

$$4\sin^2\theta = 1$$

$$\sin\theta = 1/2$$ 

($$For 0^\circ < \theta < 90^\circ,$$)

$$\theta = 30\degree$$

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

$$\frac{\cot^2 2 \times 30\degree + \sec^2 2 \times 30\degree}{\tan^2 2 \times 30\degree - \sin^2 2 \times 30\degree}$$

$$\frac{\cot^2 60\degree + \sec^2 60\degree}{\tan^2 60\degree - \sin^2 60\degree}$$

$$\frac{(\frac{1}{\sqrt3})^2 + 2^2}{(\sqrt{3})^2 - (\frac{\sqrt3}{2})^2}$$

$$\frac{\frac{1}{3} + 4}{3 - \frac{3}{4}}$$

$$\frac{\frac{13}{3} }{\frac{9}{4}}$$ = 52/27