linear algebra - Characteristic polynomial of a unitary matrix.

Given a matrix a unitary $A$ $\in$ $M_{n \times n} (\mathbb{R})$, how does one show that its characteristic polynomial $\triangle_A(t)$ satisfies the following:

$t^n \triangle_A(1/t) = \pm \triangle_A(t)$.

I see that the characteristic polynomial is essentially symmetric (or anti-symmetric). I have shown that the determinant of a unitary matrix are $\pm 1$ and that its eigenvalues all have modulus 1. I feel that there is a connection between these properties and the structure of its characteristic polynomial.

If we are dealing with real numbers, it could happen that all of the eigenvalues are either 1 or -1. Then, using the binomial theorem, we would have $\triangle_A(t)= (t \pm 1)^n$, and we would obtain a symmetric or anti-symmetric polynomial.

However, I am stuck here.