Eliminate θ between the given equations:

x = a cot θ + b cosec θ, y = a cosec θ + b cot θ

x = a cot θ + b cosec θ …..(1)

y = a cosec θ + b cot θ …..(2)

Subtract equation (2) from (1):

⇒ x - y = a cot θ + b cosec θ - (a cosec θ + b cot θ)

⇒ x - y = a cot θ + b cosec θ - a cosec θ - b cot θ

⇒ x - y = a (cot θ - cosec θ) - b (cot θ - cosec θ)

⇒ x - y = (a - b) (cot θ - cosec θ)

Add equation (2) from (1):

⇒ x + y = a cot θ + b cosec θ + (a cosec θ + b cot θ)

⇒ x + y = a (cot θ + cosec θ) + b (cot θ + cosec θ)

⇒ x + y = (a + b) (cot θ + cosec θ)

Now multiply:

⇒ (x + y)(x - y) = (a + b)(a - b)(cot θ + cosec θ)(cot θ - cosec θ)

⇒ x² - y² = (a² - b²)(cot² θ - cosec² θ)

By identity: cot² θ − cosec ² θ = −1

⇒ x² - y² = - (a² - b²)

⇒ x² - y² = b² - a²

Hence, the required relation is x² - y² = b² - a² .