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.
Best Answer
For symmetric:
$$\sin^2 x + cos ^2 y = 1$$
We need to show that:
$$\sin^2 y + \cos^2 x = 1$$
Given: $$\sin^2 x + cos ^2 y = 1$$ $$\implies 1-\cos^2 x + 1- sin ^2 y = 1$$
$$\implies \cos^2 x + \sin^2 y = 1$$