Power Reduction Formulas/Hyperbolic Cosine Cubed

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Theorem

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

where $\cosh$ denotes hyperbolic cosine.


Proof 1

\(\displaystyle \cosh 3 x\) \(=\) \(\displaystyle 4 \cosh^3 x - 3 \cosh x\) Triple Angle Formula for Hyperbolic Cosine
\(\displaystyle \implies \ \ \) \(\displaystyle 4 \cosh^3 x\) \(=\) \(\displaystyle \cosh 3 x + 3 \cosh x\) rearranging
\(\displaystyle \implies \ \ \) \(\displaystyle \cosh^3 x\) \(=\) \(\displaystyle \dfrac {\cosh 3 x + 3 \cosh x} 4\) dividing both sides by $4$

$\blacksquare$


Proof 2

\(\displaystyle \cosh^3 x\) \(=\) \(\displaystyle \frac 1 {2^3} \paren {e^x + e^{-x} }^3\) Definition of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 8 \paren {e^{3x} + e^{-3x} + 3e^{x} + 3e^{-x} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 4 \paren {\frac{ e^{3x} + e^{-3x} } 2} + \frac 3 4 \paren {\frac{e^{x} + e^{-x} } 2}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cosh 3x} 4 + \frac {3 \cosh x} 4\) Definition of Hyperbolic Cosine

$\blacksquare$


Proof 3

\(\displaystyle \cosh^3 x\) \(=\) \(\displaystyle \cos^3 i x\) Hyperbolic Cosine in terms of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\map \cos {3 i x} + 3 \cos i x} 4\) Cube of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cosh 3 x + 3 \cosh x} 4\) Hyperbolic Cosine in terms of Cosine

$\blacksquare$


Also see


Sources