Is $\{(x,y):\sin^2x+\sin^2y=0\}$ a subspace of $\mathbb{ R}^2$?
Is $\{(x,y):\sin^2x+\cos^2y=1\}$ a subspace of $\mathbb{ R}^2$?
My attempt: subspaces of $\mathbb{ R}^2$ are precisely ${0}$, $\mathbb{ R}^2$ and the lines passing through origin.Since the above sets don't constitute any of these subspaces, they are not subspaces.Is my attempt correct
The idea is good enough, but the statement that your set is equal to none of those sets without any proof is a bit too brief. Why is it not any of those subspaces? Once you've given a proof of that, your answer is good to go.
That being said, I would personally have preferred your answer had it instead had the approach of how your set breaks the axioms of vector spaces, rather than the exhaustive listing of the subspaces of $\Bbb R^2$ and showing that your set is none of them. It is how these problems are usually solved, and basically what you will have to do anyway.