How do you evaluate sec(-pi)?

$\sec \left(- \pi\right) = \frac{1}{\cos} \left(- \pi\right)$
The $\cos$ function is an even function, and that means that $\cos \left(- x\right) = \cos x$, for all values of $x$. In particular $\cos \left(- \pi\right) = \cos \pi = - 1$. Then, since $\sec \left(- \pi\right) = \frac{1}{\cos} \left(- \pi\right) = \frac{1}{- 1} = - 1$