Eliminate θ between the given equations:

x = a cosec θ, y = b cot θ

Given,

x = a cosec θ, y = b cot θ

⇒ cosec θ = $\dfrac{x}{a}$

⇒ cot θ = $\dfrac{y}{b}$

Using the identity

cosec² θ - cot² θ = 1

Substitute the values:

$\Rightarrow \Big(\dfrac{x}{a}\Big)^2 - \Big(\dfrac{y}{b}\Big)^2 = 1 \\[1em] \Rightarrow \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$

Hence, the required relation is $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$.