#### Answer

=0

#### Work Step by Step

\[\begin{gathered}
\mathop {\lim }\limits_{\theta \, \to \,0} \,\,\frac{{{{\cos }^2}\theta - 1}}{\theta } \hfill \\
\hfill \\
factoring\,\,the\,\,numerator\, \hfill \\
\hfill \\
= \mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\,\,\frac{{\,\left( {\cos \,\,\theta + 1} \right)\,\left( {\cos \theta - 1} \right)}}{\theta } \hfill \\
\hfill \\
use\,\,the\,\,product\,\,of\,\,limits\,law \hfill \\
\hfill \\
= \,\left( {\mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\,\,\left( {\cos \,\,\theta + 1} \right)} \right)\,\left( {\mathop {\lim }\limits_{\theta \, \to \,0} \,\,\frac{{cos\theta - 1}}{\theta }} \right) \hfill \\
\hfill \\
from\,\,the\,\,theorem\,\,7.11 \hfill \\
\hfill \\
\mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\,\frac{{\cos \,\,\theta - 1}}{\theta } = 0\,\, \hfill \\
\hfill \\
then \hfill \\
\hfill \\
= \,\left( {1 + 1} \right)\,\left( 0 \right) \hfill \\
\hfill \\
= 0 \hfill \\
\hfill \\
\end{gathered} \]