Define a relation R on the interval $\left[0,\frac{\mathrm{\pi }}{2}\right)$ by x R y if and only if sec²x – tan²y = 1. Then R is :

sec²x – tan²x = 1 (on replacing y with x)
$\Rightarrow$Reflexive
sec²x – tan²y = 1
$\Rightarrow$ 1 + tan²x + 1 – sec²y = 1
$\Rightarrow$ sec²y – tan²x = 1
$\Rightarrow$ symmetric
sec²x – tan²y = 1,
sec²y – tan²z = 1
Adding both
$\Rightarrow$ sec²x – tan²y + sec²y – tan²z = 1 + 1
sec²x + 1 – tan²z = 2
sec²x – tan²z = 1
$\Rightarrow$ Transitive
Hence equivalence relation