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 . x) * ((tan . x) - (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) ) ) )
A1:
for x being Real st x in Z holds
sin is_differentiable_in x
by SIN_COS:64;
assume A2:
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 . x) * ((tan . x) - (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) ) )
then A3:
Z c= (dom (tan - cot)) /\ (dom sin)
by VALUED_1:def 4;
then A4:
Z c= dom (tan - cot)
by XBOOLE_1:18;
then A5:
tan - cot is_differentiable_on Z
by Th5;
Z c= dom sin
by A3, XBOOLE_1:18;
then A6:
sin is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sin (#) (tan - cot)) `| Z) . x = ((cos . x) * ((tan . x) - (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))))
proof
let x be
Real;
( x in Z implies ((sin (#) (tan - cot)) `| Z) . x = ((cos . x) * ((tan . x) - (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) )
assume A7:
x in Z
;
((sin (#) (tan - cot)) `| Z) . x = ((cos . x) * ((tan . x) - (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))))
then ((sin (#) (tan - cot)) `| Z) . x =
(((tan - cot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((tan - cot),x)))
by A2, A5, A6, FDIFF_1:21
.=
(((tan . x) - (cot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((tan - cot),x)))
by A4, A7, VALUED_1:13
.=
(((tan . x) - (cot . x)) * (cos . x)) + ((sin . x) * (diff ((tan - cot),x)))
by SIN_COS:64
.=
(((tan . x) - (cot . x)) * (cos . x)) + ((sin . x) * (((tan - cot) `| Z) . x))
by A5, A7, FDIFF_1:def 7
.=
(((tan . x) - (cot . x)) * (cos . x)) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))))
by A4, A7, Th5
;
hence
((sin (#) (tan - cot)) `| Z) . x = ((cos . x) * ((tan . x) - (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))))
;
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 . x) * ((tan . x) - (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) ) )
by A2, A5, A6, FDIFF_1:21; verum