from typing import Callable
import numpy as np

z0 = 20


def find_root(f: Callable[[float], float], left: float, right: float, tolerance: float = 1e-6,
              max_iter: int = 100) -> float | None:
    a = f(left)
    b = f(right)
    x = None
    for _ in range(max_iter):
        if abs(a - b) < tolerance:
            break
        x = left - a * (left - right) / (a - b)
        y = f(x)
        if abs(y) < tolerance:
            break
        right, b = left, a
        left, a = x, y
    return x


def cot(x):
    return 1 / np.tan(x)


def func(z: float) -> float:
    first = np.sqrt((z0 / z) ** 2 - 1)
    second = cot(z)
    return first + second


print(find_root(func, 0.1, 3, max_iter=1000))
print(find_root(func, 3.1, 6, max_iter=1000))
print(find_root(func, 6.1, 8, max_iter=1000))
print(find_root(func, 10, 12, max_iter=1000))