\(\cos(n!)\) and \(\cos^n(n!)\)

[Grigorios Kostakos]

Examine the convergence of the following sequences: \begin{align*} {\rm{a)}} \qquad& \alpha_{n}=\cos(n!)\,,\quad n\in\mathbb{N}\,,\\ {\rm{b)}} \qquad& \beta_{n}=\cos^{n}(n!)\,,\quad n\in\mathbb{N}\,. \end{align*}



NOTE: I have no solution for both.