Proof
We consider the
cosine series
-
For
,
we have
.
For
,
one can write
Hence, due to
the intermediate value theorem,
there exists at least one zero in the given interval.
To prove uniqueness, we consider the
derivative
of cosine, which is
-
due to
fact.
Hence, it is enough to show that sine is positive in the interval , because then cosine is
strictly decreasing
by
fact
in the interval and there is only one zero. Now, for
,
we have