Prove along the following path that x and y must be

Question

We know, or else, it is not hard to prove, that if X and Y are independent standard normal random variables, then X + Y and X - Y are independent. This problem is devoted to a converse. Suppose that X and Y are independent symmetric random variables with mean 0 and variance 1, such that X + Y and X - Y are independent. Let ? denote the common characteristic function of X and Y. Prove, along the following path, that X and Y must be standard normal random variables:

(a) Prove that ?(2t)=(?(t))⁴.

(b) Iterate.