theorem :: FDIFF_7:31

for n being Element of NAT
for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (cos ^))) & n > 0 & ( for x being Real st x in Z holds
cos . x <> 0 ) holds
( (1 / n) (#) ((#Z n) * (cos ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) ) )