# Secant Function is Even

## Theorem

Let $x \in \R$ be a real number.

Let $\sec x$ be the secant of $x$.

Then, whenever $\sec x$ is defined:

$\sec \left({-x}\right) = \sec x$

That is, the secant function is even.

## Proof

 $\displaystyle \sec \left({-x}\right)$ $=$ $\displaystyle \frac 1 {\cos \left({-x}\right)}$ Secant is Reciprocal of Cosine $\displaystyle$ $=$ $\displaystyle \frac 1 {\cos x}$ Cosine Function is Even $\displaystyle$ $=$ $\displaystyle \sec x$ Secant is Reciprocal of Cosine

$\blacksquare$