Answer
\[ - \frac{{2\sqrt 3 }}{3}\]
Work Step by Step
\[\begin{gathered}
\sec \left( {\frac{{7\pi }}{6}} \right) \hfill \\
{\text{Write }}\frac{{7\pi }}{6}{\text{ as }}\pi + \frac{\pi }{6} \hfill \\
\sec \left( {\frac{{7\pi }}{6}} \right) = \sec \left( {\pi + \frac{\pi }{6}} \right) \hfill \\
{\text{Use the identity }}\sec \theta = \frac{1}{{\cos \theta }} \hfill \\
\sec \left( {\pi + \frac{\pi }{6}} \right) = \frac{1}{{\cos \left( {\pi + \frac{\pi }{6}} \right)}} \hfill \\
{\text{Where cos}}\left( {A + B} \right) = \cos A\cos B - \sin A\sin B \hfill \\
\sec \left( {\frac{{7\pi }}{6}} \right) = \frac{1}{{\cos \left( \pi \right)\cos \left( {\frac{\pi }{6}} \right) - \sin \left( \pi \right)\sin \left( {\frac{\pi }{6}} \right)}} \hfill \\
\sec \left( {\frac{{7\pi }}{6}} \right) = \frac{1}{{\left( { - 1} \right){\frac{{\sqrt 3 }}{2}} }} \hfill \\
\sec \left( {\frac{{7\pi }}{6}} \right) = - \frac{2}{{\sqrt 3 }} \hfill \\
\sec \left( {\frac{{7\pi }}{6}} \right) = - \frac{{2\sqrt 3 }}{3} \hfill \\
\end{gathered} \]
