let Z be open Subset of REAL; :: thesis: ( Z c= dom cosec & Z c= ].(- 1),1.[ implies ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) )
assume that
A1: Z c= dom cosec and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
for x being Real st x in Z holds
cosec is_differentiable_in x
proof
let x be Real; :: thesis: ( x in Z implies cosec is_differentiable_in x )
assume x in Z ; :: thesis: cosec is_differentiable_in x
then sin . x <> 0 by A1, RFUNCT_1:3;
hence cosec is_differentiable_in x by FDIFF_9:2; :: thesis: verum
end;
then A4: cosec is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then Z c= dom arctan by A2, XBOOLE_1:1;
then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan + arccot) by VALUED_1:def 1;
then Z c= (dom cosec) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (cosec (#) (arctan + arccot)) by VALUED_1:def 4;
for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) )
assume A9: x in Z ; :: thesis: ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))
then A10: sin . x <> 0 by A1, RFUNCT_1:3;
((cosec (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def 1
.= (((arctan . x) + (arccot . x)) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_9:2
.= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def 7
.= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * 0) by A2, A9, Th37
.= - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ;
hence ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ; :: thesis: verum
end;
hence ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) by A8, A4, A3, FDIFF_1:21; :: thesis: verum