An identity checker

You will write three small functions that check a claimed identity on many angles: one finds the worst miss between the two sides, one says whether the identity holds, and one finds an angle where it fails.

The idea. For a true identity, the two sides agree to within rounding error, about $10^{-16}$. A tolerance of $10^{-9}$ is safe. For a false identity, some angle shows a miss far bigger than that.

Your tasks.

1. In worst_miss, return the largest absolute difference between the two arrays.
2. In holds, return True when the two arrays agree everywhere within 1e-9, and False otherwise.
3. In first_counterexample, return the first angle where the sides differ by more than 1e-9, or None when there is none.