Define a relation R on the interval $$[0,\frac{\pi}{2})$$ by $$xRy$$ if and only if$$\sec^{2} x-\tan^{2} y=1$$. Then R is :

We use the identity $$\sec^2 \theta = 1 + \tan^2 \theta$$. The condition becomes:

$$(1 + \tan^2 x) - \tan^2 y = 1 \implies \tan^2 x = \tan^2 y$$

• Reflexive: $$\tan^2 x = \tan^2 x$$ is always true. (Reflexive)

• Symmetric: If $$\tan^2 x = \tan^2 y$$, then $$\tan^2 y = \tan^2 x$$. (Symmetric)

• Transitive: If $$\tan^2 x = \tan^2 y$$ and $$\tan^2 y = \tan^2 z$$, then $$\tan^2 x = \tan^2 z$$. (Transitive)

Since it satisfies all three, $$R$$ is an equivalence relation.

Correct Option: B