let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( tan + cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) ) ) )

assume A1: Z c= dom () ; :: thesis: ( tan + cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) ) )

then A2: Z c= () /\ () 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
proof end;
then A4: tan is_differentiable_on Z by ;
A5: Z c= dom cot by ;
for x being Real st x in Z holds
cot is_differentiable_in x
proof end;
then A6: cot is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) )
assume A7: x in Z ; :: thesis: (() `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))
then A8: sin . x <> 0 by ;
A9: cos . x <> 0 by ;
(() `| Z) . x = (diff (tan,x)) + (diff (cot,x)) by
.= (1 / ((cos . x) ^2)) + (diff (cot,x)) by
.= (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) by ;
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
(() `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) ) ) by ; :: thesis: verum