If $$5 \cos^{2} \theta + 1 = 3 \sin^{2} \theta, 0^\circ < \theta < 90^\circ$$, then what is the value of $$\frac{\tan \theta + \sec \theta}{\cot \theta + \cosec \theta}$$

$$5 \cos^{2} \theta + 1 = 3 \sin^{2} \theta$$

$$5 \cos^{2} \theta + \sin^{2} \theta + \cos^{2} \theta = 3 \sin^{2} \theta$$

$$6 \cos^{2} \theta = 2 \sin^{2} \theta$$

$$\tan^{2} \theta = 3$$

$$\tan \theta = \sqrt3$$ 
$$\theta = 60\degree$$

$$\frac{\tan \theta + \sec \theta}{\cot \theta + \cosec \theta}$$

= $$\frac{\tan 60\degree + \sec 60\degree}{\cot 60\degree + \cosec 60\degree}$$

= $$\frac{\sqrt{3} + 2}{\frac{1}{\sqrt3} + \frac{2}{\sqrt3}}$$

= $$\frac{\sqrt{3} + 2}{\sqrt3}$$

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