### If ^@ x^@ = ^@a{\space} cosec{\space} \theta^@ and ^@y^@ = ^@b{\space} sec{\space} \theta^@, prove that ^@\bigg ( \dfrac { a^2 } { x^2 } + \dfrac { b^2 } { y^2 } \bigg)^@ = 1.

**Answer:**

1

**Step by Step Explanation:**

- We are told that @^ \begin{aligned} & x = a{\space} cosec{\space} \theta \\ {\implies} & \dfrac {a} {x} = \dfrac { 1 } { cosec {\space} \theta } \\ {\implies} & \dfrac {a} {x} = sin {\space} \theta &&\ldots \text{(i)} \\ \end{aligned} @^ Also, @^ \begin{aligned} & y = b {\space} sec{\space} \theta \\ {\implies} & \dfrac {b} {y} = \dfrac { 1 } { sec {\space} \theta } \\ {\implies} & \dfrac {b} {y} = cos {\space} \theta &&\ldots \text{(ii)} \end{aligned} @^
- On squaring and adding ^@ \text{(i)} ^@ and ^@ \text{(ii)}^@, we get @^ \bigg ( \dfrac { a^2 } { x^2 } + \dfrac { b^2 } { y^2 } \bigg) = sin^2{\space} \theta + cos^2 {\space} \theta = 1 @^