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

$$3 \sec^2 \theta + \tan \theta = 7$$

$$3 (1 + tan^2\theta) + \tan \theta = 7$$

$$4tan^2\theta = 4$$

$$tan^2\theta = 1$$

$$\theta = 45\degree$$

$$\frac{\cosec 2 \theta + \cos \theta}{\sin 2 \theta + \cot \theta}$$

= $$\frac{\cosec 2 \times 45\degree + \cos 45\degree}{\sin 2 \times 45\degree + \cot 45\degree}$$

= $$\frac{\cosec 90\degree + \cos 45\degree}{\sin 90\degree + \cot 45\degree}$$

Put the value of $$\theta$$,

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

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

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

= $$\frac{\sqrt{2} + 2}{4}$$