Power Reduction Formulas/Hyperbolic Cosine to 4th
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Theorem
- $\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$
where $\cos$ denotes hyperbolic cosine.
Proof 1
\(\ds \cosh 4 x\) | \(=\) | \(\ds \paren {\cosh^2 x}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {\cosh 2 x + 1} 2}^2\) | Square of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh^2 2 x + 2 \cosh 2 x + 1} 4\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac {\cosh 4 x + 1} 2 + 2 \cosh 2 x + 1} 4\) | Square of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh 4 x + 1 + 4 \cosh 2 x + 2} 8\) | multiplying top and bottom by $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cosh 2 x + \cosh 4 x} 8\) | rearrangement |
$\blacksquare$
Proof 2
\(\ds \cosh^4 x\) | \(=\) | \(\ds \frac 1 {2^4}\left(e^{x} + e^{-x}\right)^4\) | Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {16} \left({e^{4 x} + 4 e^{2 x} + 6 e^{0 x} + 4 e^{-2 x} + e^{-4 x} }\right)\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \left({\frac{e^{4 x} + e^{-4 x} } 2}\right) + \frac 4 8 \left({\frac{e^{2 x} + e^{-2 x} } 2 }\right) + \frac 6 {16}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cosh 2 x + \cosh 4 x} 8\) | Definition of Hyperbolic Cosine |
$\blacksquare$
Proof 3
\(\ds \cosh^4 x\) | \(=\) | \(\ds \cos^4 i x\) | Hyperbolic Cosine in terms of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cos \paren {2 i x} + \cos \paren {4 i x} } 8\) | Fourth Power of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cosh 2 x + \cosh 4 x} 8\) | Hyperbolic Cosine in terms of Cosine |
$\blacksquare$
Also see
- Square of Hyperbolic Sine
- Square of Hyperbolic Cosine
- Cube of Hyperbolic Sine
- Cube of Hyperbolic Cosine
- Fourth Power of Hyperbolic Sine
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.41$: Powers of Hyperbolic Functions