Hyperbolic Secant Function is Even/Proof 2

Theorem

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

Proof

\(\ds \sech \paren {-x}\)

\(=\)

\(\ds \frac 1 {\cosh \paren {-x} }\)

Definition of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos \paren {-i x} }\)

Hyperbolic Cosine in terms of Cosine

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos \paren {i x} }\)

Cosine Function is Even

\(\ds \)

\(=\)

\(\ds \frac 1 {\cosh x}\)

Hyperbolic Cosine in terms of Cosine

\(\ds \)

\(=\)

\(\ds \sech x\)

Definition of Hyperbolic Secant

$\blacksquare$