If $$\frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta} = \frac{3}{5}$$, then the value of $$\frac{\cosec \theta + \cot \theta}{\cosec \theta - \cot \theta}$$ is:

$$\frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta} = \frac{3}{5}$$

Divide numerator and den. by tan$$\theta$$,

 $$\frac{\cosec \theta - 1}{\cosec \theta + 1} = \frac{3}{5}$$

Using componendo dividendo rule,

(if $$\frac{p + q}{p - q} = \frac{a}{b} then \frac{p}{q} = \frac{a + b}{a - b}$$)

cosec$$\theta = \frac{5 + 3}{5 - 3} = 4

$$cot^2θ=cosec^2θ−1$$

$$\therefore \cot^2 \theta = 4^2 - 1$$

$$\therefore \cot \theta = \sqrt {15}$$

$$\frac{cosec \theta + cot \theta}{cosec \theta - cot \theta} = \frac{4 + \sqrt{15}}{4 - \sqrt{15}}​​$$

Rationalizing, we get

= $$\frac{4 + \sqrt {15}}{4 - \sqrt {15}} \times \frac{4 + \sqrt {15}}{4 + \sqrt {15}}$$

= 31 + 8$$\sqrt{15}$$