theorem :: FDIFF_7:30

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