Answer
\begin{align*}
\frac{d^2 }{dt^2} \cos^2 t&=-2\cos 2t .
\end{align*}
Work Step by Step
We have
\begin{align*}
\frac{d}{dt} \cos^2 t&=2\cos t (-\sin t)\\
&=-2\sin t \cos t\\
&=-\sin 2t
\end{align*}
and hence
\begin{align*}
\frac{d^2 }{dt^2} \cos^2 t&=-\frac{d}{dt} \sin 2 t\\
&=-2\cos 2t .
\end{align*}
