Define a relation \(R\) on the interval \(\left[0, \frac{\pi}{2}\right)\) by \(x R y\) if and only if \(\sec ^2 x-\tan ^…

Given \(x R y \Leftrightarrow \sec ^2 x-\tan ^2 y=1\)

Using \(\sec ^2 x-1=\tan ^2 x\), we get \(\tan ^2 x=\tan ^2 y\)

Since \(x, y \in\left[0, \frac{\pi}{2}\right)\),

we have \(\tan x \geq 0, \tan y \geq 0\), so \(\tan x=\tan y\)

Now \(\tan x\) is one-one on \(\left[0, \frac{\pi}{2}\right)\), hence \(x=y\)

So the relation becomes \(x R y \Longleftrightarrow x=y\)

Therefore \(R\) is reflexive, symmetric and transitive