let Z be open Subset of REAL; ( Z c= dom ((- cot) - cosec) & ( for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) implies ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) ) )
assume that
A1:
Z c= dom ((- cot) - cosec)
and
A2:
for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 )
; ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) )
Z c= (dom (- cot)) /\ (dom (sin ^))
by A1, VALUED_1:12;
then A3:
Z c= dom (- cot)
by XBOOLE_1:18;
then A4:
Z c= dom cot
by VALUED_1:8;
for x being Real st x in Z holds
cot is_differentiable_in x
then A5:
cot is_differentiable_on Z
by A4, FDIFF_1:9;
then A6:
(- 1) (#) cot is_differentiable_on Z
by A3, FDIFF_1:20;
A7:
for x being Real st x in Z holds
sin . x <> 0
by A4, FDIFF_8:2;
then A8:
sin ^ is_differentiable_on Z
by FDIFF_4:40;
for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x))
proof
let x be
Real;
( x in Z implies (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) )
assume A9:
x in Z
;
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x))
then A10:
sin . x <> 0
by A4, FDIFF_8:2;
A11:
1
+ (cos . x) <> 0
by A2, A9;
(((- cot) - cosec) `| Z) . x =
(diff ((- cot),x)) - (diff ((sin ^),x))
by A1, A8, A6, A9, FDIFF_1:19
.=
((((- 1) (#) cot) `| Z) . x) - (diff ((sin ^),x))
by A6, A9, FDIFF_1:def 7
.=
((- 1) * (diff (cot,x))) - (diff ((sin ^),x))
by A3, A5, A9, FDIFF_1:20
.=
((- 1) * (- (1 / ((sin . x) ^2)))) - (diff ((sin ^),x))
by A10, FDIFF_7:47
.=
(1 / ((sin . x) ^2)) - (((sin ^) `| Z) . x)
by A8, A9, FDIFF_1:def 7
.=
(1 / ((sin . x) ^2)) - (- ((cos . x) / ((sin . x) ^2)))
by A7, A9, FDIFF_4:40
.=
(1 / ((sin . x) ^2)) + ((cos . x) / ((sin . x) ^2))
.=
(1 + (cos . x)) / ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2))
by XCMPLX_1:62
.=
(1 + (cos . x)) / (1 - ((cos . x) ^2))
by SIN_COS:28
.=
(1 + (cos . x)) / ((1 + (cos . x)) * (1 - (cos . x)))
.=
((1 + (cos . x)) / (1 + (cos . x))) / (1 - (cos . x))
by XCMPLX_1:78
.=
1
/ (1 - (cos . x))
by A11, XCMPLX_1:60
;
hence
(((- cot) - cosec) `| Z) . x = 1
/ (1 - (cos . x))
;
verum
end;
hence
( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) )
by A1, A8, A6, FDIFF_1:19; verum