If x = acosecθ + b cotθ, y = bcosecθ + a cot θ, then what will be the value of $$\text{x}^2-\text{y}^2$$​ ?

Correct option is A

Given:

$$x = a \cosec\theta + b \cot\theta$$​​
$$y = b \cosec\theta + a \cot\theta$$​​

Formula Used:

​$$\cosec^2\theta - \cot^2\theta = 1$$

x² - y² = (x - y) (x + y) 

Solution:

​$$x + y = (a + b)\cosec\theta + (a + b)\cot\theta = (a + b)(\cosec\theta + \cot\theta)$$ ​

​$$x - y = (a - b)\cosec\theta + (b - a)\cot\theta = (a - b)(\cosec\theta - \cot\theta)$$​

​$$x^2 - y^2 = (a + b)(\cosec\theta + \cot\theta) \times (a - b)(\cosec\theta - \cot\theta)$$​

So,

​$$x^2 - y^2 = (a^2 - b^2)(\cosec^2\theta - \cot^2\theta)$$​

​$$x^2 - y^2 = (a^2 - b^2) \times 1 = a^2 - b^2$$​