An exact series expansion for the Dottie number

In this article, an exact series expansion for the Dottie number (solution of the equation $\cos (x)=x$) is presented. Its derivation consists in combining the Kaplan representation of the Dottie number as a series in odd powers of $\pi$, with a series for ${\pi }^{2k+1}$ ($k\ge 1$) obtained using the Faà di Bruno formula.