let Z be open Subset of REAL; ( Z c= dom (sin * (tan - cot)) implies ( sin * (tan - cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (tan - cot)) `| Z) . x = (cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) ) ) )
assume A1:
Z c= dom (sin * (tan - cot))
; ( sin * (tan - cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (tan - cot)) `| Z) . x = (cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) ) )
dom (sin * (tan - cot)) c= dom (tan - cot)
by RELAT_1:25;
then A2:
Z c= dom (tan - cot)
by A1, XBOOLE_1:1;
then A3:
tan - cot is_differentiable_on Z
by Th5;
A4:
for x being Real st x in Z holds
sin * (tan - cot) is_differentiable_in x
then A6:
sin * (tan - cot) is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sin * (tan - cot)) `| Z) . x = (cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))
proof
let x be
Real;
( x in Z implies ((sin * (tan - cot)) `| Z) . x = (cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) )
A7:
sin is_differentiable_in (tan - cot) . x
by SIN_COS:64;
assume A8:
x in Z
;
((sin * (tan - cot)) `| Z) . x = (cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))
then
tan - cot is_differentiable_in x
by A3, FDIFF_1:9;
then diff (
(sin * (tan - cot)),
x) =
(diff (sin,((tan - cot) . x))) * (diff ((tan - cot),x))
by A7, FDIFF_2:13
.=
(cos . ((tan - cot) . x)) * (diff ((tan - cot),x))
by SIN_COS:64
.=
(cos . ((tan - cot) . x)) * (((tan - cot) `| Z) . x)
by A3, A8, FDIFF_1:def 7
.=
(cos . ((tan - cot) . x)) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))
by A2, A8, Th5
.=
(cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))
by A2, A8, VALUED_1:13
;
hence
((sin * (tan - cot)) `| Z) . x = (cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))
by A6, A8, FDIFF_1:def 7;
verum
end;
hence
( sin * (tan - cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (tan - cot)) `| Z) . x = (cos . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) ) )
by A1, A4, FDIFF_1:9; verum