Given P(E1 U E2) = 1, P(E1) = P(E2), P(E1 intersection E2) = P(E2) * P(E2) (since E1 & E2 are independent).
Now by using inclusion exclusion principle, P(E1 U E2) = P(E1) + P(E2) - P(E1 intersection E2).
Let P(E1) = a, then P(E2) = a, and P(E1 intersection E2) = a * a. Putting all these values along with P(E1 U E2) = 1, leads to a quadratic equation a^2 - 2a + 1.
This equation gives a = P(E1) = P(E2) = 1.