Power Reduction Formulas/Hyperbolic Sine to 4th
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Theorem
- $\sinh^4 x = \dfrac {3 - 4 \cosh 2 x + \cosh 4 x} 8$
where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.
Proof
\(\ds \sinh^4 x\) | \(=\) | \(\ds \left({\sinh^2 x}\right)^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({\frac {\cosh 2 x - 1} 2}\right)^2\) | Square of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh^2 2 x - 2 \cosh 2 x + 1} 4\) | 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 |
$\blacksquare$
Also see
- Square of Hyperbolic Sine
- Square of Hyperbolic Cosine
- Cube of Hyperbolic Sine
- Cube of Hyperbolic Cosine
- Fourth Power of Hyperbolic Cosine
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.40$: Powers of Hyperbolic Functions