Double Angle Formulas/Hyperbolic Sine/Proof 2

Theorem

$\sinh 2 x = 2 \sinh x \cosh x$

Proof

\(\ds \sinh 2 x\)

\(=\)

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

Definition of Hyperbolic Sine

\(\ds \)

\(=\)

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

Difference of Two Squares

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac{e^x + e^{-x} } 2 \cdot \frac {e^x - e^{-x} } 2}\)

\(\ds \)

\(=\)

\(\ds 2 \sinh x \cosh x\)

Definition of Hyperbolic Sine, Definition of Hyperbolic Cosine

$\blacksquare$