[FOM] question

[Sergei Akbarov]

Dear colleagues,

Could you, please, answer one question, concerning elementary
mathematics? I am a specialist in functional analysis, so 
please, forgive me, for my incompetence in such problems.

My question is as follows. Suppose we have an elementary
function $f(x)$ of one variable $x$ (I use term "elementary" in the
same sense as it is used in school, i.e. $f(x)$ must be a
finite combination of functions like $x^a$, $a^x$, $\sin x$, $+$, $-$,
$\cdot$, $/$, etc.) And suppose $f(x)$ is constant on some interval
$I$. Does it mean that $x$ can be eliminated from $f(x)$ by using a
finite list of elementary identities, like $\sin^2 x+\cos^2 x=1$?

Thank you in advance,
Sergei Akbarov,
Moscow