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:

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

That is, the secant function is even.


Proof

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

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.32$
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
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