let Z be open Subset of REAL; ( Z c= dom (((id Z) - tan) - sec) & ( for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) implies ( ((id Z) - tan) - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) ) )
assume that
A1:
Z c= dom (((id Z) - tan) - sec)
and
A2:
for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 )
; ( ((id Z) - tan) - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) )
A3:
Z c= (dom ((id Z) - tan)) /\ (dom sec)
by A1, VALUED_1:12;
then A4:
Z c= dom ((id Z) - tan)
by XBOOLE_1:18;
then A5:
Z c= (dom (id Z)) /\ (dom tan)
by VALUED_1:12;
then A6:
Z c= dom tan
by XBOOLE_1:18;
A7:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:18;
A8:
Z c= dom (id Z)
by A5, XBOOLE_1:18;
then A9:
id Z is_differentiable_on Z
by A7, FDIFF_1:23;
for x being Real st x in Z holds
tan is_differentiable_in x
then A10:
tan is_differentiable_on Z
by A6, FDIFF_1:9;
then A11:
(id Z) - tan is_differentiable_on Z
by A4, A9, FDIFF_1:19;
A12:
Z c= dom sec
by A3, XBOOLE_1:18;
then A13:
sec is_differentiable_on Z
by FDIFF_9:4;
A14:
for x being Real st x in Z holds
(((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2))
proof
let x be
Real;
( x in Z implies (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) )
assume A15:
x in Z
;
(((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2))
then A16:
cos . x <> 0
by A6, FDIFF_8:1;
then A17:
(cos . x) ^2 > 0
by SQUARE_1:12;
(((id Z) - tan) `| Z) . x =
(diff ((id Z),x)) - (diff (tan,x))
by A4, A10, A9, A15, FDIFF_1:19
.=
(((id Z) `| Z) . x) - (diff (tan,x))
by A9, A15, FDIFF_1:def 7
.=
1
- (diff (tan,x))
by A8, A7, A15, FDIFF_1:23
.=
1
- (1 / ((cos . x) ^2))
by A16, FDIFF_7:46
.=
1
- ((((cos . x) ^2) + ((sin . x) ^2)) / ((cos . x) ^2))
by SIN_COS:28
.=
1
- ((((cos . x) ^2) / ((cos . x) ^2)) + (((sin . x) ^2) / ((cos . x) ^2)))
by XCMPLX_1:62
.=
1
- (1 + (((sin . x) ^2) / ((cos . x) ^2)))
by A17, XCMPLX_1:60
.=
- (((sin . x) ^2) / ((cos . x) ^2))
;
hence
(((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2))
;
verum
end;
for x being Real st x in Z holds
((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1)
proof
let x be
Real;
( x in Z implies ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) )
assume A18:
x in Z
;
((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1)
then A19:
1
+ (sin . x) <> 0
by A2;
((((id Z) - tan) - sec) `| Z) . x =
(diff (((id Z) - tan),x)) - (diff (sec,x))
by A1, A13, A11, A18, FDIFF_1:19
.=
((((id Z) - tan) `| Z) . x) - (diff (sec,x))
by A11, A18, FDIFF_1:def 7
.=
(- (((sin . x) ^2) / ((cos . x) ^2))) - (diff (sec,x))
by A14, A18
.=
(- (((sin . x) ^2) / ((cos . x) ^2))) - ((sec `| Z) . x)
by A13, A18, FDIFF_1:def 7
.=
(- (((sin . x) ^2) / ((cos . x) ^2))) - ((sin . x) / ((cos . x) ^2))
by A12, A18, FDIFF_9:4
.=
- (((sin . x) / ((cos . x) ^2)) + (((sin . x) ^2) / ((cos . x) ^2)))
.=
- (((sin . x) + ((sin . x) ^2)) / ((cos . x) ^2))
by XCMPLX_1:62
.=
- (((sin . x) * (1 + (sin . x))) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2)))
.=
- (((sin . x) * (1 + (sin . x))) / (1 - ((sin . x) ^2)))
by SIN_COS:28
.=
- (((sin . x) * (1 + (sin . x))) / ((1 + (sin . x)) * (1 - (sin . x))))
.=
- ((((sin . x) * (1 + (sin . x))) / (1 + (sin . x))) / (1 - (sin . x)))
by XCMPLX_1:78
.=
- (((sin . x) * ((1 + (sin . x)) / (1 + (sin . x)))) / (1 - (sin . x)))
by XCMPLX_1:74
.=
- (((sin . x) * 1) / (1 - (sin . x)))
by A19, XCMPLX_1:60
.=
(sin . x) / (- (1 - (sin . x)))
by XCMPLX_1:188
.=
(sin . x) / ((sin . x) - 1)
;
hence
((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1)
;
verum
end;
hence
( ((id Z) - tan) - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) )
by A1, A13, A11, FDIFF_1:19; verum