$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian.

Question

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian.

Without loss of generality we may assume that $X$ and $Y$ are centralized and standardized. My attempts are given as follows:

1. I tried to use characteristic functions first. Let $\varphi(t)$ denotes the characteristic function of $X$(and of $Y$ as well). Then we can deduce from the assumptions that for any $t_1, t_2 \in \mathbb{R}$, $$ \varphi(t_1 + t_2)\varphi(t_1 - t_2) = [\varphi(t_1)]^2\varphi(t_2)\varphi(-t_2) $$ Then I tried to conclude by L'Hopital's theorem that $\varphi'(t) = -t\varphi(t)$ but failed.
2. Then I tried to compute the moments(as Gaussian r.v. is determined by its moments) but got nothing desired.

Would you please give me some hints on this problem? Thanks in advance for any help!