Power Reduction Formulas/Hyperbolic Sine Squared/Proof 2

Theorem

$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$

Proof

\(\ds \sinh^2 x\)

\(=\)

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

Definition of Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \frac 1 4 \paren {e^{2 x} + e^{-2 x} - 2}\)

multiplying out

\(\ds \)

\(=\)

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

rearranging

\(\ds \)

\(=\)

\(\ds \frac {\cosh 2 x - 1} 2\)

Definition of Hyperbolic Cosine

$\blacksquare$