If I have two random variables $X$ and $Y$ which are independent and identically distributed (i.i.d) as Standard normal $\sim N(0,1)$
does $E(X^2) = E(XY)$ since $X$ and $Y$ are i.i.d?
please explain it step by step, I'm a beginner in probability. and if it is true, is it true for any pair of i.i.d random variable no matter what distribution it is (provided that it has a finite mean)?
No.
Since $X$ has variance $1$ and expectation $0$ $$\mathbb{E}[X^2] =\mathbb{E}[X^2] - \mathbb{E}[X]^2 = \operatorname{Var}[X] = 1$$ by definition, while, since $X,Y$ are independent,
$$ \mathbb{E}[XY] = \mathbb{E}[X]\mathbb{E}[Y]=0\cdot 0 = 0\,. $$