let Z be open Subset of REAL; ( Z c= dom (tan (#) sec) implies ( tan (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) sec) `| Z) . x = ((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2)) ) ) )
assume A1:
Z c= dom (tan (#) sec)
; ( tan (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) sec) `| Z) . x = ((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2)) ) )
then A2:
Z c= (dom tan) /\ (dom sec)
by VALUED_1:def 4;
then A3:
Z c= dom tan
by XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x
then A4:
tan is_differentiable_on Z
by A3, FDIFF_1:9;
A5:
Z c= dom sec
by A2, XBOOLE_1:18;
A6:
for x being Real st x in Z holds
( sec is_differentiable_in x & diff (sec,x) = (sin . x) / ((cos . x) ^2) )
then
for x being Real st x in Z holds
sec is_differentiable_in x
;
then A7:
sec is_differentiable_on Z
by A5, FDIFF_1:9;
for x being Real st x in Z holds
((tan (#) sec) `| Z) . x = ((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2))
proof
let x be
Real;
( x in Z implies ((tan (#) sec) `| Z) . x = ((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2)) )
assume A8:
x in Z
;
((tan (#) sec) `| Z) . x = ((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2))
then A9:
cos . x <> 0
by A3, FDIFF_8:1;
((tan (#) sec) `| Z) . x =
((sec . x) * (diff (tan,x))) + ((tan . x) * (diff (sec,x)))
by A1, A4, A7, A8, FDIFF_1:21
.=
((sec . x) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff (sec,x)))
by A9, FDIFF_7:46
.=
((sec . x) * (1 / ((cos . x) ^2))) + ((tan . x) * ((sin . x) / ((cos . x) ^2)))
by A6, A8
.=
((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2))
by A5, A8, RFUNCT_1:def 2
;
hence
((tan (#) sec) `| Z) . x = ((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2))
;
verum
end;
hence
( tan (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) sec) `| Z) . x = ((1 / ((cos . x) ^2)) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2)) ) )
by A1, A4, A7, FDIFF_1:21; verum