Hyperbolic Cosine minus Hyperbolic Cosine/Proof 2

Theorem

$\cosh x - \cosh y = 2 \map \sinh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$

Proof

\(\ds 2 \map \sinh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}\)

\(=\)

\(\ds \dfrac 1 2 \paren {e^{\frac {x + y} 2} - e^{-\frac {x + y} 2} } \paren {e^{\frac {x - y} 2} - e^{-\frac {x - y} 2} }\)

Definition of Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren {e^x - e^y - e^{-y} + e^{-x} }\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {e^x + e^{-x} } 2 - \dfrac {e^y + e^{-y} } 2\)

rearranging

\(\ds \)

\(=\)

\(\ds \cosh x - \cosh y\)

Definition of Hyperbolic Cosine

$\blacksquare$