theorem :: FDIFF_5:9

for Z being open Subset of REAL st Z c= dom ((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) & not 0 in Z holds
( (sin * ((id Z) ^)) (#) (cos * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2)) ) )