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

Let $\theta = \cot^{-1}y$

This means $\cot\theta = y$

Using the identity $1 + \cot^2\theta = \csc^2\theta$:

$\csc^2\theta = 1 + \cot^2\theta$

$\csc^2\theta = 1 + y^2$

Therefore:

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

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

---

Let $\alpha = \tan^{-1}x$

This means $\tan\alpha = x$

Using the identity $1 + \tan^2\alpha = \sec^2\alpha$:

$\sec^2\alpha = 1 + \tan^2\alpha$

$\sec^2\alpha = 1 + x^2$

Therefore:

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

---

Combining both terms:

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

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

$= x^2 - y^2$

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