let Z be open Subset of REAL; ( Z c= dom (cot (#) cosec) implies ( cot (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) cosec) `| Z) . x = (- ((1 / ((sin . x) ^2)) / (sin . x))) - (((cot . x) * (cos . x)) / ((sin . x) ^2)) ) ) )
assume A1:
Z c= dom (cot (#) cosec)
; ( cot (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) cosec) `| Z) . x = (- ((1 / ((sin . x) ^2)) / (sin . x))) - (((cot . x) * (cos . x)) / ((sin . x) ^2)) ) )
then A2:
Z c= (dom cot) /\ (dom cosec)
by VALUED_1:def 4;
then A3:
Z c= dom cot
by XBOOLE_1:18;
for x being Real st x in Z holds
cot is_differentiable_in x
then A4:
cot is_differentiable_on Z
by A3, FDIFF_1:9;
A5:
Z c= dom cosec
by A2, XBOOLE_1:18;
A6:
for x being Real st x in Z holds
( cosec is_differentiable_in x & diff (cosec,x) = - ((cos . x) / ((sin . x) ^2)) )
then
for x being Real st x in Z holds
cosec is_differentiable_in x
;
then A7:
cosec is_differentiable_on Z
by A5, FDIFF_1:9;
for x being Real st x in Z holds
((cot (#) cosec) `| Z) . x = (- ((1 / ((sin . x) ^2)) / (sin . x))) - (((cot . x) * (cos . x)) / ((sin . x) ^2))
proof
let x be
Real;
( x in Z implies ((cot (#) cosec) `| Z) . x = (- ((1 / ((sin . x) ^2)) / (sin . x))) - (((cot . x) * (cos . x)) / ((sin . x) ^2)) )
assume A8:
x in Z
;
((cot (#) cosec) `| Z) . x = (- ((1 / ((sin . x) ^2)) / (sin . x))) - (((cot . x) * (cos . x)) / ((sin . x) ^2))
then A9:
sin . x <> 0
by A3, FDIFF_8:2;
((cot (#) cosec) `| Z) . x =
((cosec . x) * (diff (cot,x))) + ((cot . x) * (diff (cosec,x)))
by A1, A4, A7, A8, FDIFF_1:21
.=
((cosec . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff (cosec,x)))
by A9, FDIFF_7:47
.=
((cosec . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (- ((cos . x) / ((sin . x) ^2))))
by A6, A8
.=
((- (1 / ((sin . x) ^2))) / (sin . x)) + ((cot . x) * (- ((cos . x) / ((sin . x) ^2))))
by A5, A8, RFUNCT_1:def 2
;
hence
((cot (#) cosec) `| Z) . x = (- ((1 / ((sin . x) ^2)) / (sin . x))) - (((cot . x) * (cos . x)) / ((sin . x) ^2))
;
verum
end;
hence
( cot (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) cosec) `| Z) . x = (- ((1 / ((sin . x) ^2)) / (sin . x))) - (((cot . x) * (cos . x)) / ((sin . x) ^2)) ) )
by A1, A4, A7, FDIFF_1:21; verum