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:def 1;
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 (id Z)
by XBOOLE_1:18;
A7:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:35;
then A8:
id Z is_differentiable_on Z
by A6, FDIFF_1:31;
A9:
Z c= dom tan
by A5, XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x
then A10:
tan is_differentiable_on Z
by A9, FDIFF_1:16;
then A11:
(id Z) - tan is_differentiable_on Z
by A4, A8, FDIFF_1:27;
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 A9, FDIFF_8:1;
then A17:
(cos . x) ^2 > 0
by SQUARE_1:74;
(((id Z) - tan ) `| Z) . x =
(diff (id Z),x) - (diff tan ,x)
by A4, A10, A8, A15, FDIFF_1:27
.=
(((id Z) `| Z) . x) - (diff tan ,x)
by A8, A15, FDIFF_1:def 8
.=
1
- (diff tan ,x)
by A6, A7, A15, FDIFF_1:31
.=
1
- (1 / ((cos . x) ^2 ))
by A16, FDIFF_7:46
.=
1
- ((((cos . x) ^2 ) + ((sin . x) ^2 )) / ((cos . x) ^2 ))
by SIN_COS:31
.=
1
- ((((cos . x) ^2 ) / ((cos . x) ^2 )) + (((sin . x) ^2 ) / ((cos . x) ^2 )))
by XCMPLX_1:63
.=
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_1:26
.=
((((id Z) - tan ) `| Z) . x) + (diff sec ,x)
by A11, A18, FDIFF_1:def 8
.=
(- (((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 8
.=
(- (((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) * (sin . x))) / ((cos . x) ^2 )
by XCMPLX_1:121
.=
((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:31
.=
((sin . x) * (1 - (sin . x))) / ((1 - (sin . x)) * (1 + (sin . x)))
.=
(((sin . x) * (1 - (sin . x))) / (1 - (sin . x))) / (1 + (sin . x))
by XCMPLX_1:79
.=
((sin . x) * ((1 - (sin . x)) / (1 - (sin . x)))) / (1 + (sin . x))
by XCMPLX_1:75
.=
((sin . x) * 1) / (1 + (sin . x))
by A19, XCMPLX_1:60
.=
(sin . x) / (1 + (sin . x))
;
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_1:26; verum