import math


def my_iteration(n, x0, y0):
    func1 = lambda x, y: math.tan(x * y + 0.5) - x ** 2
    deriv1x = lambda x, y: y / (math.cos(x * y + 0.5) ** 2) - 2 * x
    deriv1y = lambda x, y: x / (math.cos(x * y + 0.5) ** 2)
    func2 = lambda x, y: x ** 2 + y ** 2 - 1
    deriv2x = lambda x, y: 2 * x
    deriv2y = lambda x, y: 2 * y
    matrix = [[0, 0], [0, 0]]
    with open("output.txt", "w") as file:
        for i in range(0, n):
            matrix[0][0] = deriv2y(x0, y0)
            matrix[1][0] = -deriv2x(x0, y0)
            matrix[0][1] = -deriv1y(x0, y0)
            matrix[1][1] = deriv1x(x0, y0)
            det = matrix[0][0] * matrix[1][1] - matrix[1][0] * matrix[0][1]
            for k in range(2):
                for j in range(2):
                    matrix[k][j] /= det
            f = [0, 0]
            f[0] = func1(x0, y0)
            f[1] = func2(x0, y0)
            x0 -= matrix[0][0] * f[0] + matrix[0][1] * f[1]
            y0 -= matrix[1][0] * f[0] + matrix[1][1] * f[1]
            file.write(str(i) + " : " + str(x0) + " " + str(y0) + "\n")


my_iteration(20, 1, 1)