The value of $-cosec^2(\cot^{-1}y) + \sec^2 (\tan^{-1}x)$ is equal to

$x^2 - y^2$

The correct answer is Option (2) → $x^2 - y^2$

Given expression:

$- \csc^2(\cot^{-1}y) + \sec^2(\tan^{-1}x)$

Using identities:

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

$\csc^2(\cot^{-1}y) = 1 + y^2$

Substitute:

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

$= -1 - y^2 + 1 + x^2$

$= x^2 - y^2$

Therefore, the value is $x^2 - y^2$.