Answer
\[ = - \sec x\]
Work Step by Step
\[\begin{gathered}
Using\,\,the\,trigonometric\,identity\,for\,the\,cosine\,of\,a\,sum \hfill \\
\cos \,\,\left( {a + b} \right) = \cos a\cos b - \sin a\sin b \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
\sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( {x + \pi } \right)}} \hfill \\
\hfill \\
{\text{Simplify}} \hfill \\
\hfill \\
\sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( x \right)\cos \,\left( \pi \right) - \sin \,\left( x \right)\sin \,\left( \pi \right)}} \hfill \\
\hfill \\
\sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( x \right) \cdot \,\left( { - 1} \right) - \sin \,\left( x \right) \cdot 0}} = \frac{1}{{ - \cos \,\left( x \right)}} = - \sec x \hfill \\
\hfill \\
which\,was\,needed\,to\,be\,shown. \hfill \\
\end{gathered} \]
