let Z be open Subset of REAL ; :: thesis: ( 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 ) ; :: thesis: ( 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 Z c= (dom tan ) /\ (dom cot ) by VALUED_1:def 1;
then A2: ( Z c= dom tan & Z c= dom cot ) by XBOOLE_1:18;
for x being Real st x in Z holds
cot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cot is_differentiable_in x )
assume x in Z ; :: thesis: cot is_differentiable_in x
then sin . x <> 0 by A2, FDIFF_8:2;
hence cot is_differentiable_in x by FDIFF_7:47; :: thesis: verum

end;

then A3: cot is_differentiable_on Z by A2, FDIFF_1:16;
for x being Real st x in Z holds
tan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan is_differentiable_in x )
assume x in Z ; :: thesis: tan is_differentiable_in x
then cos . x <> 0 by A2, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A4: tan is_differentiable_on Z by A2, FDIFF_1:16;
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; :: thesis: ( x in Z implies ((tan + cot ) `| Z) . x = (1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )) )
assume A5: x in Z ; :: thesis: ((tan + cot ) `| Z) . x = (1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 ))
then A6: ( sin . x <> 0 & cos . x <> 0 ) by A2, FDIFF_8:1, FDIFF_8:2;
((tan + cot ) `| Z) . x = (diff tan ,x) + (diff cot ,x) by A1, A3, A4, A5, FDIFF_1:26
.= (1 / ((cos . x) ^2 )) + (diff cot ,x) by A6, FDIFF_7:46
.= (1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )) by A6, FDIFF_7:47 ;
hence ((tan + cot ) `| Z) . x = (1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )) ; :: thesis: 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, A3, A4, FDIFF_1:26; :: thesis: verum