Power Reduction Formulas/Hyperbolic Cosine Cubed/Proof 2

Theorem

$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$

Proof

\(\ds \cosh^3 x\)

\(=\)

\(\ds \frac 1 {2^3} \paren {e^x + e^{-x} }^3\)

Definition of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \frac 1 8 \paren {e^{3x} + e^{-3x} + 3e^{x} + 3e^{-x} }\)

\(\ds \)

\(=\)

\(\ds \frac 1 4 \paren {\frac{ e^{3x} + e^{-3x} } 2} + \frac 3 4 \paren {\frac{e^{x} + e^{-x} } 2}\)

\(\ds \)

\(=\)

\(\ds \frac {\cosh 3x} 4 + \frac {3 \cosh x} 4\)

Definition of Hyperbolic Cosine

$\blacksquare$