#!/usr/bin/env python3
"""Numerical illustration, not an empirical experiment or the analytic proof."""
from math import cos, pi


def check():
    errors = [abs(cos(i * .05 * pi / 180) ** 2 - (1 + cos(2 * i * .05 * pi / 180)) / 2) for i in range(3601)]
    maximum = max(errors)
    assert maximum < 1e-12, maximum
    print(f'3,601 angles, 0–180 degrees; maximum absolute discrepancy: {maximum:.3g}')
    print('Analytic proof: cos(2θ) = 2cos²(θ) − 1. Numerical differences are roundoff.')
    return maximum


if __name__ == '__main__':
    check()