Show that random variables $X$ and $Y$ are not independent, but nevertheless Cov$[X,Y] = 0$

It is not necessary to find these functions.

To prove dependency it is enough to find sets $A,B$ such that $$P(X\in A\wedge Y\in B)\neq P(X\in A)P(Y\in B)$$

To prove that the covariance is $0$ it is enough to show that $$\mathbb EXY=\mathbb EX\mathbb EY$$

and for that you do not need the PDF's either.

E.g. note that: $$\mathbb EXY=\int_0^1\sin2\pi z\cos2\pi z~\mathrm dz$$