Secant Function is Even

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Theorem

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

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


Then, whenever $\sec x$ is defined:

$\map \sec {-x} = \sec x$

That is, the secant function is even.


Proof

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

$\blacksquare$


Also see


Sources