def NonRec(x, eps):
    s=1
    term=1
    i=0
    while (abs(term) > eps):
        term=term*(x/(i+1))
        s+=term
        i+=1
    return s

import math

def cosine(x, eps=1e-5):
    res = 1
    i = 1
    x = x * math.pi / 180.
    while True:
        delta = (x**(2*i) / math.factorial(2*i))
        res += (-1)**i * delta
        i += 1
        if delta <= eps:
            return res, i

In [50]: res, n = cosine(123, eps=1e-6)

In [51]: print(res, n)
-0.544639044595552 8