Power Reduction Formulas/Hyperbolic Sine to 4th

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.

$\ds \sinh^4 x$

$=$

$\ds \left({\sinh^2 x}\right)^2$

$\ds $

$=$

$\ds \left({\frac {\cosh 2 x - 1} 2}\right)^2$

$\ds $

$=$

$\ds \frac {\cosh^2 2 x - 2 \cosh 2 x + 1} 4$

multiplying out

$\ds $

$=$

$\ds \frac {\frac {\cosh 4 x + 1} 2 - 2 \cosh 2 x + 1} 4$

$\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

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.40$: Powers of Hyperbolic Functions