Is E(X|f(Y)) same as E(X|Y)? I am trying to see whether the conditional expectaction of E(X|Y) is the same as conditional expectaction of E(X|f(Y)), where f(Y) is any function of Y which is smooth, e.g. polynomial, sin etc.

It's not. Take a trivial example, where $Y$ is a random variable that takes the $\{-1,0,1\}$ values, and $X = Y + \varepsilon$. You would see different results if you look at $E[X|Y]$ vs $E[X|f(Y)]$ where $f(y) = y^2$, because in the second case, the right-hand side is equal to $1$ maps to "two times more" $X$'s: the cases where for the raw data $Y=1$ and $Y=-1$.
