If tan θ = $\Big(\dfrac{1}{\sqrt{7}}\Big)$, then the value of $\Big(\dfrac{\cosec^2\theta + \sec^2\theta}{\cosec^2\theta - \sec^2\theta}\Big)$ is:

1. $\Big(\dfrac{3}{4}\Big)$

2. $\Big(\dfrac{4}{3}\Big)$

3. $\Big(\dfrac{3}{7}\Big)$

4. $\Big(\dfrac{4}{7}\Big)$

Given,

tan θ = $\Big(\dfrac{1}{\sqrt{7}}\Big)$

tan² θ = $\Big(\dfrac{1}{7}\Big)$

We know that,

$\sec^2 \theta = 1 + \tan^2 \theta \\[1em] \sec^2 \theta = 1 + \dfrac{1}{7} \\[1em] \sec^2 \theta = \dfrac{8}{7}. \\[1em] \cosec^2 \theta = 1 + \cot^2 \theta \\[1em] \cosec^2 \theta = 1 + 7 \\[1em] \cosec^2 \theta = 8.$

Given expression,

$\Rightarrow \Big(\dfrac{\cosec^2\theta + \sec^2\theta}{\cosec^2\theta - \sec^2\theta}\Big) \\[1em] \Rightarrow \dfrac{8 + \dfrac{8}{7}}{8 - \dfrac{8}{7}} \\[1em] \Rightarrow \dfrac{\dfrac{56 + 8}{7}}{\dfrac{56 - 8}{7}} \\[1em] \Rightarrow \dfrac{64}{48} = \dfrac{4}{3}.$

Hence, option 2 is the correct option.