Secant in terms of Hyperbolic Secant

Theorem

Let $z \in \C$ be a complex number.

Then:

$\sec z = \map \sech {i z}$

where:

$\sec$ denotes the secant function

$\sech$ denotes the hyperbolic secant

$i$ is the imaginary unit: $i^2 = -1$.

Proof

\(\ds \sec z\)

\(=\)

\(\ds \frac 1 {\cos z}\)

Definition of Complex Secant Function

\(\ds \)

\(=\)

\(\ds \frac 1 {\map \cosh {i z} }\)

Cosine in terms of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \map \sech {i z}\)

Definition of Hyperbolic Secant

$\blacksquare$

Also see

• Sine in terms of Hyperbolic Sine
• Cosine in terms of Hyperbolic Cosine
• Tangent in terms of Hyperbolic Tangent
• Cotangent in terms of Hyperbolic Cotangent
• Cosecant in terms of Hyperbolic Cosecant

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.84$: Relationship between Hyperbolic and Trigonometric Functions