I think your teacher assumes you already know the characteristic function of a Gaussian random variable: if , then . This is well-known but not quite easy to prove, you can't find it by direct computation. One way is to see that, if and , then (you need to differentiate and then integrate by part), and , hence . The case follows. But, once again: you can't be expected to invent that, this is the kind of thing one is supposed to know (not the proof, but the formula).