Power Reduction Formulas/Hyperbolic Sine Cubed

Theorem

$\sinh^3 x = \dfrac {\sinh 3 x - 3 \sinh x} 4$

where $\sin$ denotes hyperbolic sine.

$\ds \sinh 3 x$

$=$

$\ds 3 \sinh x + 4 \sinh^3 x$

$\ds \leadsto \ \ $

$\ds 4 \sinh^3 x$

$=$

$\ds \sinh 3 x - 3 \sinh x$

rearranging

$\ds \leadsto \ \ $

$\ds \sinh^3 x$

$=$

$\ds \frac {\sinh 3 x - 3 \sinh x} 4$

dividing both sides by $4$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.38$: Powers of Hyperbolic Functions