Determine whether $R$ is an equivalence relation: $xRy$ if $\cos(x)^2+\sin(y)^2=1$ I'm having troubles with this question. I understand that for a relation to be equivalent, it needs to be reflexive, symmetric, and transitive.
So far I've split this problem into 3 sections, one to check each requirement.
But am I overthinking it?  Can I just state that:
Using Pythagoras' theorem, $((\cos(x))^2)+((\sin(x))^2)=1$, so we see that $x = y$. So $x$ is related to $y$ by equality, therefore this relation is an equivalence relation.

You cannot say that, because $x,y$ may be different values; that is to say, we need not have
$$\cos^2(x) + \sin^2(y) = 1$$
As an example, take $x = 0, y = \pi/2$. Then $\cos(x) = \sin(y) = 1$ and thus the sum of their squares is $2$.

Granted, this argument does work if $x=y$, i.e. if we're showing $xRx$. But you cannot conclude $R$ is an equivalence relation from that alone.
