let Z be open Subset of REAL; ( Z c= dom (cos * (tan + cot)) implies ( cos * (tan + cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (tan + cot)) `| Z) . x = - ((sin . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))) ) ) )
assume A1:
Z c= dom (cos * (tan + cot))
; ( cos * (tan + cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (tan + cot)) `| Z) . x = - ((sin . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))) ) )
dom (cos * (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 Th6;
A4:
for x being Real st x in Z holds
cos * (tan + cot) is_differentiable_in x
then A6:
cos * (tan + cot) is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cos * (tan + cot)) `| Z) . x = - ((sin . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))))
proof
let x be
Real;
( x in Z implies ((cos * (tan + cot)) `| Z) . x = - ((sin . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))) )
A7:
cos is_differentiable_in (tan + cot) . x
by SIN_COS:63;
assume A8:
x in Z
;
((cos * (tan + cot)) `| Z) . x = - ((sin . ((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 (
(cos * (tan + cot)),
x) =
(diff (cos,((tan + cot) . x))) * (diff ((tan + cot),x))
by A7, FDIFF_2:13
.=
(- (sin . ((tan + cot) . x))) * (diff ((tan + cot),x))
by SIN_COS:63
.=
(- (sin . ((tan + cot) . x))) * (((tan + cot) `| Z) . x)
by A3, A8, FDIFF_1:def 7
.=
(- (sin . ((tan + cot) . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))
by A2, A8, Th6
.=
(- (sin . ((tan . x) + (cot . x)))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))
by A2, A8, VALUED_1:def 1
;
hence
((cos * (tan + cot)) `| Z) . x = - ((sin . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))))
by A6, A8, FDIFF_1:def 7;
verum
end;
hence
( cos * (tan + cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (tan + cot)) `| Z) . x = - ((sin . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))) ) )
by A1, A4, FDIFF_1:9; verum