I feel a little confused about varieties.
If I am asked to determine whether V(cos(x),sin(x)) seen in R1 is a variety, would it be true to just say it is?, because the polynomial 0=1 has no solution (since we know that cos(x) and sin(x) don't coincide at their zeros), so V(cos(x),sin(x))=R.
But if I am now being asked to determine whether V((cos(x),sin(x))) seen in R2 is a variety (if this makes sense) what can I say?. I think it should not be a variety but when trying to see how may I integrate polynomials into my thinking, I get confused. What reasoning should I be using for this case?