let Z be open Subset of REAL; ( Z c= dom (sin (#) tan) implies ( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) tan) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2)) ) ) )
A1:
for x being Real st x in Z holds
sin is_differentiable_in x
by SIN_COS:64;
assume A2:
Z c= dom (sin (#) tan)
; ( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) tan) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2)) ) )
then A3:
Z c= (dom sin) /\ (dom tan)
by VALUED_1:def 4;
then A4:
Z c= dom tan
by XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x
then A5:
tan is_differentiable_on Z
by A4, FDIFF_1:9;
Z c= dom sin
by A3, XBOOLE_1:18;
then A6:
sin is_differentiable_on Z
by A1, FDIFF_1:9;
A7:
for x being Real st x in Z holds
diff (tan,x) = 1 / ((cos . x) ^2)
for x being Real st x in Z holds
((sin (#) tan) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2))
proof
let x be
Real;
( x in Z implies ((sin (#) tan) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2)) )
assume A8:
x in Z
;
((sin (#) tan) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2))
then ((sin (#) tan) `| Z) . x =
((diff (sin,x)) * (tan . x)) + ((sin . x) * (diff (tan,x)))
by A2, A5, A6, FDIFF_1:21
.=
((cos . x) * (tan . x)) + ((sin . x) * (diff (tan,x)))
by SIN_COS:64
.=
((cos . x) * (tan . x)) + ((sin . x) * (1 / ((cos . x) ^2)))
by A7, A8
.=
(((sin . x) / (cos . x)) * ((cos . x) / 1)) + ((sin . x) / ((cos . x) ^2))
by A4, A8, RFUNCT_1:def 1
.=
((sin . x) * ((cos . x) / (cos . x))) + ((sin . x) / ((cos . x) ^2))
.=
((sin . x) * 1) + ((sin . x) / ((cos . x) ^2))
by A4, A8, FDIFF_8:1, XCMPLX_1:60
.=
(sin . x) + ((sin . x) / ((cos . x) ^2))
;
hence
((sin (#) tan) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2))
;
verum
end;
hence
( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) tan) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2)) ) )
by A2, A5, A6, FDIFF_1:21; verum