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

$$7 \sin^2 \theta - \cos^2 \theta + 2 \sin \theta = 2$$

$$\Rightarrow$$ $$7 \sin^{2} \theta - \cos^{2} \theta + 2 \sin \theta - 2 = 0$$

$$\Rightarrow$$ $$7 \sin^{2} \theta - (1 - \sin^{2} \theta) + 2 \sin \theta - 2 = 0$$

$$\Rightarrow$$ $$8 \sin^{2} \theta + 2 \sin \theta - 3 = 0$$

$$\Rightarrow$$ $$8 \sin^{2} \theta + 6 \sin \theta - 4\sin \theta- 3 = 0$$

$$\Rightarrow$$ $$2\sin \theta(4 \sin\theta + 3) -1(4 \sin\theta + 3) = 0$$

$$\Rightarrow$$ $$(2\sin \theta - 1)(4 \sin\theta + 3) = 0$$

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

$$\sin \theta = 1/2$$

$$\theta = 30\degree$$

$$\frac{\sec 2 \times 30+ \cot 2\times 30}{\cosec 2\times 30 + \tan 2\times 30}$$

$$\Rightarrow$$ $$\frac{\sec 60 + \cot 60}{\cosec 60 + \tan 60}$$ 
$$\Rightarrow$$ $$\frac{2+ \frac{1}{\sqrt3}}{\frac{2}{\sqrt3} + \sqrt3}$$

$$\Rightarrow$$ $$\frac{2\sqrt3 + 1}{2+ 3}$$

$$\Rightarrow$$ $$\frac{2\sqrt3 + 1}{5}$$