There are many problems with your code.
A comma is missing in the NDSolve, between {alpha[t], beta[t]} and {t, 0, 0.000001}.
You are using Abs, which is not differentiable in the complex domain. You should use RealAbs, which is differentiable in the real domain outside the origin.
Your initial condition alpha[0] == 0 happens to be precisely in the point where RealAbs[Tan[alpha[t]]] is not differentiable.
