Answer
$$\eqalign{
& {\text{Decreasing on }}\left( { - \pi , - \frac{\pi }{2}} \right),{\mkern 1mu} \left( {0,\frac{\pi }{2}} \right) \cr
& {\text{Increasing on }}\left( { - \frac{\pi }{2},0} \right),\left( {\frac{\pi }{2},\pi } \right) \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {\cos ^2}x{\text{ on the interval }}\left[ { - \pi ,\pi } \right] \cr
& {\text{Differentiate the function}} \cr
& f'\left( x \right) = 2\left( {\cos x} \right)\left( { - \sin x} \right) \cr
& f'\left( x \right) = - \sin 2x \cr
& {\text{Set the derivative to 0}} \cr
& - \sin 2x = 0 \cr
& {\text{Solving the equation for the interval }}\left[ { - \pi ,\pi } \right]{\text{ we obtain}} \cr
& x = - \pi ,{\mkern 1mu} {\mkern 1mu} - \frac{\pi }{2},{\mkern 1mu} {\mkern 1mu} 0,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{\pi }{2},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \pi \cr
& {\text{From the critical values we can make the next intervals}}\, \cr
& \left( { - \pi , - \frac{\pi }{2}} \right),{\mkern 1mu} {\mkern 1mu} \left( { - \frac{\pi }{2},0} \right),{\mkern 1mu} {\mkern 1mu} \left( {0,\frac{\pi }{2}} \right),{\mkern 1mu} {\mkern 1mu} \left( {\frac{\pi }{2},\pi } \right) \cr
& {\text{Now, we will evaluate the critical value and resume in a table}} \cr} $$
\[\begin{array}{*{20}{c}}
{{\text{Interval}}}&{{\text{Test value}}\left( x \right)}&{{\text{Sign of }}f'\left( x \right)}&{{\text{Behavior of }}f\left( x \right)} \\
{\left( { - \pi , - \frac{\pi }{2}} \right)}&{ - \frac{{5\pi }}{6}}& - &{{\text{Decreasing}}} \\
{\left( { - \frac{\pi }{2},0} \right)}&{ - \frac{\pi }{4}}& + &{{\text{Increasing}}} \\
{\left( {0,\frac{\pi }{2}} \right)}&{\frac{\pi }{4}}& - &{{\text{Decreasing}}} \\
{\left( {\frac{\pi }{2},\pi } \right)}&{\frac{{5\pi }}{6}}& + &{{\text{Increasing}}} \\
{}&{}&{}&{} \\
{}&{}&{}&{}
\end{array}\]
$$\eqalign{
& {\text{From the table we can conlude that the function is:}} \cr
& {\text{Decreasing on }}\left( { - \pi , - \frac{\pi }{2}} \right),{\mkern 1mu} \left( {0,\frac{\pi }{2}} \right) \cr
& {\text{Increasing on }}\left( { - \frac{\pi }{2},0} \right),\left( {\frac{\pi }{2},\pi } \right) \cr} $$
