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

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

A2: for x being Real st x in Z holds
cos . x <> 0
proof
let x be Real; :: thesis: ( x in Z implies cos . x <> 0 )
assume x in Z ; :: thesis:
then x in dom () by ;
hence cos . x <> 0 by FDIFF_8:1; :: thesis: verum
end;
A3: for x being Real st x in Z holds
sin * tan is_differentiable_in x
proof end;
then A5: sin * tan is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = (cos (tan . x)) / ((cos . x) ^2)
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (cos (tan . x)) / ((cos . x) ^2) )
A6: sin is_differentiable_in tan . x by SIN_COS:64;
assume A7: x in Z ; :: thesis: (() `| Z) . x = (cos (tan . x)) / ((cos . x) ^2)
then A8: cos . x <> 0 by A2;
then tan is_differentiable_in x by FDIFF_7:46;
then diff ((),x) = (diff (sin,(tan . x))) * (diff (tan,x)) by
.= (cos (tan . x)) * (diff (tan,x)) by SIN_COS:64
.= (cos (tan . x)) * (1 / ((cos . x) ^2)) by
.= (cos (tan . x)) / ((cos . x) ^2) ;
hence ((sin * tan) `| Z) . x = (cos (tan . x)) / ((cos . x) ^2) by ; :: thesis: verum
end;
hence ( sin * tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (cos (tan . x)) / ((cos . x) ^2) ) ) by ; :: thesis: verum