let Z be open Subset of REAL; ( Z c= dom (tan - sec) & ( for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) implies ( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) ) )
assume that
A1:
Z c= dom (tan - sec)
and
A2:
for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 )
; ( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) )
Z c= (dom tan) /\ (dom (cos ^))
by A1, VALUED_1:12;
then A3:
Z c= dom tan
by XBOOLE_1:18;
then A4:
for x being Real st x in Z holds
cos . x <> 0
by FDIFF_8:1;
then A5:
cos ^ is_differentiable_on Z
by FDIFF_4:39;
for x being Real st x in Z holds
tan is_differentiable_in x
then A6:
tan is_differentiable_on Z
by A3, FDIFF_1:9;
for x being Real st x in Z holds
((tan - sec) `| Z) . x = 1 / (1 + (sin . x))
proof
let x be
Real;
( x in Z implies ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) )
assume A7:
x in Z
;
((tan - sec) `| Z) . x = 1 / (1 + (sin . x))
then A8:
cos . x <> 0
by A3, FDIFF_8:1;
A9:
1
- (sin . x) <> 0
by A2, A7;
((tan - sec) `| Z) . x =
(diff (tan,x)) - (diff ((cos ^),x))
by A1, A5, A6, A7, FDIFF_1:19
.=
(1 / ((cos . x) ^2)) - (diff ((cos ^),x))
by A8, FDIFF_7:46
.=
(1 / ((cos . x) ^2)) - (((cos ^) `| Z) . x)
by A5, A7, FDIFF_1:def 7
.=
(1 / ((cos . x) ^2)) - ((sin . x) / ((cos . x) ^2))
by A4, A7, FDIFF_4:39
.=
(1 - (sin . x)) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2))
by XCMPLX_1:120
.=
(1 - (sin . x)) / (1 - ((sin . x) ^2))
by SIN_COS:28
.=
(1 - (sin . x)) / ((1 + (sin . x)) * (1 - (sin . x)))
.=
((1 - (sin . x)) / (1 - (sin . x))) / (1 + (sin . x))
by XCMPLX_1:78
.=
1
/ (1 + (sin . x))
by A9, XCMPLX_1:60
;
hence
((tan - sec) `| Z) . x = 1
/ (1 + (sin . x))
;
verum
end;
hence
( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) )
by A1, A5, A6, FDIFF_1:19; verum