If $$5 \sin \theta = 4$$, then the value of $$\frac{\sec \theta + 4 \cot \theta}{4 \tan \theta - 5 \cos \theta}$$ is:

$$5 \sin \theta = 4$$

$$\sin \theta = 4/5$$

$$\frac{perpendicular}{hypotenuses} = \frac{4}{5}$$

By triplet 3-4-5,

Base = 3

$$cos\theta = base/hypotenuses = 3/5$$

$$tan\theta = perpendicular/base = 4/3$$

$$\frac{\sec \theta + 4 \cot \theta}{4 \tan \theta - 5 \cos \theta}$$

= $$\frac{\frac{1}{\cos \theta} +  \frac{4}{\tan \theta}}{4 \tan \theta - 5 \cos \theta}$$

= $$\frac{\frac{1}{3/5} + \frac{4}{4/3}}{4 \times 4/3 - 5 \times 3/5}$$

= $$\frac{\frac{5}{3} + 3}{4 \times 4/3 - 5 \times 3/5}$$

= $$\frac{\frac{14}{3}}{\frac{16}{3} - 3}$$

= $$\frac{14}{7}$$ = 2