For any real numbers x and y, we define x R y if and only if cosec^2x…

The correct answer is Option (4) → an equivalence relation

$cosec^2x=1+\cot^2y$

$⇒cosec^2y=cosec^2x$

$|cosec^2y|=|cosec^2x|$

for all x ∈ domain of $cosec\, x$

$|cosec\, x|=|cosec\, x|$ ⇒ Reflexive

for every $(x, y)∈R$

$|cosec\, x|=|cosec\, y|⇒(y,x)∈R$ ⇒ Symmetric

for $(x,y)∈R,(y,z)∈R$

$|cosec\, x|=|cosec\, y|, |cosec\, y|=|cosec\, z|$

$⇒|cosec\, x|=|cosec\, z|⇒(x,z)∈R$ ⇒ Transitive

⇒ R is equivalence relation