Virtually all material discussing Kepler's Equation of Elliptical Motion resolves its solution to solving the following equation numerically:
$$ E - \textit{e}\;\sin(E) = M $$
where E is the Eccentric Anomaly,
e is the eccentricity,
and M is the Mean Anomaly.
In all the material I have read and sought out, never has a solution been stated in terms of cosine.
Other than the naive approach of simply writing the existing equation as $$ E - \textit{e}\;\sqrt{1 -\cos^{2}(E)} = M $$
is there an equation corresponding to the first one--but in terms of cosine?