let Z be open Subset of REAL; ( Z c= dom (tan + cot) implies ( tan + cot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) ) ) )
assume A1:
Z c= dom (tan + cot)
; ( tan + cot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) ) )
then A2:
Z c= (dom tan) /\ (dom cot)
by VALUED_1:def 1;
then A3:
Z c= dom tan
by XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x
then A4:
tan is_differentiable_on Z
by A3, FDIFF_1:9;
A5:
Z c= dom cot
by A2, XBOOLE_1:18;
for x being Real st x in Z holds
cot is_differentiable_in x
then A6:
cot is_differentiable_on Z
by A5, FDIFF_1:9;
for x being Real st x in Z holds
((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))
proof
let x be
Real;
( x in Z implies ((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) )
assume A7:
x in Z
;
((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))
then A8:
sin . x <> 0
by A5, FDIFF_8:2;
A9:
cos . x <> 0
by A3, A7, FDIFF_8:1;
((tan + cot) `| Z) . x =
(diff (tan,x)) + (diff (cot,x))
by A1, A6, A4, A7, FDIFF_1:18
.=
(1 / ((cos . x) ^2)) + (diff (cot,x))
by A9, FDIFF_7:46
.=
(1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))
by A8, FDIFF_7:47
;
hence
((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))
;
verum
end;
hence
( tan + cot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) ) )
by A1, A6, A4, FDIFF_1:18; verum