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

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

Let \(\tan^{-1}x = \theta\), so \(\tan\theta = x\). By the Pythagorean identity \(\sec^2\theta = 1 + \tan^2\theta\):

    \(\sec^2(\tan^{-1}x) = 1 + x^2\)

Similarly, let \(\cot^{-1}y = \phi\), so \(\cot\phi = y\). By the identity \(\mathrm{cosec}^2\phi = 1 + \cot^2\phi\):

    \(\mathrm{cosec}^2(\cot^{-1}y) = 1 + y^2\)

So

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