If $ x=2 \sec ^{2} \theta $ and $ y=2 \tan ^{2} \theta-1 $, then find $ x-y $.

Given:

\( x=2 \sec ^{2} \theta \) and \( y=2 \tan ^{2} \theta-1 \).

To do:

We have to find \( x-y \).

Solution:

We know that,

$\sec ^{2} \theta - \tan ^{2} \theta = 1$

Therefore,

 $x-y=2 \sec ^{2} \theta -(2 \tan ^{2} \theta-1)$

$x-y=2 \sec ^{2} \theta -2 \tan ^{2} \theta+1$

$x-y=2(\sec ^{2} \theta - \tan ^{2} \theta)+1$

$x-y=2(1)+1$

$x-y=3$

The value of $x-y$ is $3$.