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
\(\ds \map \sec {-x}\) | \(=\) | \(\ds \frac 1 {\map \cos {-x} }\) | 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
- Sine Function is Odd
- Cosine Function is Even
- Tangent Function is Odd
- Cotangent Function is Odd
- Cosecant Function is Odd
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