If sec θ = 2x and y tan θ = 2, then the value of $2\Big(x^2 - \dfrac{1}{y^2}\Big)$ is:

1. $\Big(\dfrac{1}{2}\Big)$

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

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

4. 1

sec θ = 2x

x = $\dfrac{\sec \theta}{2}$

y tan θ = 2

$\dfrac{1}{y} = \dfrac{\tan \theta}{2}$

We have,

$\Rightarrow 2\Big(x^2 - \dfrac{1}{y^2}\Big) \\[1em] \Rightarrow 2\Big(\dfrac{\sec^2 \theta}{4} - \dfrac{\tan^2 \theta}{4}\Big) \\[1em] \Rightarrow \dfrac{2}{4}\Big(\sec^2 \theta - \tan^2 \theta\Big) \\[1em] \Rightarrow \dfrac{1}{2}\Big(\sec^2 \theta - \tan^2 \theta\Big) \\[1em] \Rightarrow \dfrac{1}{2}.$

Hence, option 1 is the correct option.