Well, you can use Complex Analysis, and the fact that since the integrand is positive, the desired integral is bounded by the improper one.
Perhaps more simple is to note the following :
(1- cos 2t)^2 / t^2 = 4 sin^2 t / t^2 < 4 min(1, 1/t^2) ( by using sin^2 t <= t^2, and 0 <= sin^2 t <= 1).
So you can bound the integrand by some function f, such that:
1. f = 4, on some interval around 0,
2. f(t) = 4/t^2 outside of said interval.
Now, compute the integral of f over -pi to pi