 Solution

Let $h=x-\pi /3$. At the expansion point, the function and its first three derivatives have values $1/2$, $-\sqrt{3}/2$, $-1/2$ and $\sqrt{3}/2$. Taylor's theorem therefore givesFor real $x$, the magnitude of the omitted fourth-order remainder is at most $|h{|}^4/24$, because the fourth derivative is another cosine and has magnitude at most one.