Normally Distributed and Covariance is zero

Question

Let $X \sim N(0,1)$, and let $P(Y=1)=P(Y=-1)=\frac{1}{2}$. Assume $X$ and $Y$ are independent, and let $Z = XY$. Prove that $Z \sim N(0,1)$ and $Cov(X,Z)=0$

For proving $Z$ is a standard normal variable, I thought using MGF would be a good idea, but could not relate that to $Z = XY$ and given conditions for variable $Y$.

Answer 1

$$f_{XY}(x,y)=\frac{1}{2}\phi(x)+\frac{1}{2}\phi(-x)=\phi(x)$$

For symmetry

For the covariance easy use the definition

$$Cov(X,Z)=E(X\cdot XY)-E(X)E(Z)=0$$

(it's trivial)