<s>[INST] Proving $\cos^2x+\sin^2y=1$ is reflexive, symmetric, transitive. I want to make sure that I got the hang of the following relations.
For reflexivity, if $x=y,\cos^2x+\sin^2y=\cos^2x+\sin^2x=1 \implies xRx$, then it is reflexive.
For symmetry, $xRy\implies\cos^2x+\sin^2y=1$ and $yRx\implies\cos^2 y+\sin^2x=1 \implies$ symmetry.
For transitivity, $\cos^2x+\sin^2y=1$ and $\cos^2y+\sin^2z=1$ then $\cos^2x+\sin^2y+\cos^2y+\sin^2z=2 \implies \cos^2x+\sin^2z=2-1=1$ so transitivity holds.
Is this enough to prove the symmetric property? Anti-symmetric is easy because I only need to prove $x=y$ but symmetry needs to be for all $x,y$ but I can't list all possibilities in all questions.
 [/INST] Reflexive
$cos^2x+sin^2x=1$
symmetric
Suppose that $cos^2x+sin^2y=1$, $cos^2y+sin^2x=1-sin^2y+1-cos^2x=2-(cos^2x+sin^2y)=2-1=1$
Transitive
$cos^2x+sin^2y=1, cos^2y+sin^2z=1$
$cos^2x+sin^2z=cos^2x+1-cos^2y=cos^2x+sin^2y=1$
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