let Z be open Subset of REAL; ( Z c= dom (cosec (#) arctan) & Z c= ].(- 1),1.[ implies ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) )
assume that
A1:
Z c= dom (cosec (#) arctan)
and
A2:
Z c= ].(- 1),1.[
; ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) )
A3:
arctan is_differentiable_on Z
by A2, SIN_COS9:81;
Z c= (dom cosec) /\ (dom arctan)
by A1, VALUED_1:def 4;
then A4:
Z c= dom cosec
by XBOOLE_1:18;
for x being Real st x in Z holds
cosec is_differentiable_in x
then A5:
cosec is_differentiable_on Z
by A4, FDIFF_1:9;
for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))
proof
let x be
Real;
( x in Z implies ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) )
assume A6:
x in Z
;
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))
then A7:
sin . x <> 0
by A4, RFUNCT_1:3;
((cosec (#) arctan) `| Z) . x =
((arctan . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arctan,x)))
by A1, A5, A3, A6, FDIFF_1:21
.=
((arctan . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arctan,x)))
by A7, FDIFF_9:2
.=
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arctan `| Z) . x))
by A3, A6, FDIFF_1:def 7
.=
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * (1 / (1 + (x ^2))))
by A2, A6, SIN_COS9:81
.=
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((1 / (sin . x)) * (1 / (1 + (x ^2))))
by A4, A6, RFUNCT_1:def 2
.=
(- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))
by XCMPLX_1:102
;
hence
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))
;
verum
end;
hence
( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) )
by A1, A5, A3, FDIFF_1:21; verum