Quadruple Angle Formulas/Hyperbolic Cosine

Theorem

$\cosh 4 x = 8 \cosh^4 x - 8 \cosh^2 x + 1$

where $\cosh$ denotes hyperbolic cosine.

$\ds \cosh 4 x$

$=$

$\ds \map \cosh {2 x + 2 x}$

$\ds $

$=$

$\ds \cosh 2 x \cosh 2 x + \sinh 2 x \sinh 2 x$

$\ds $

$=$

$\ds \paren {\cosh^2 x + \sinh^2 x} \paren {\cosh^2 x + \sinh^2 x} + \paren {2 \sinh x \cosh x} \paren {2 \sinh x \cosh x}$

$\ds $

$=$

$\ds \cosh^4 x + 2 \cosh^2 x \sinh^2 x + \sinh^4 x + 4 \cosh^2 x \sinh^2 x$

multiplying out

$\ds $

$=$

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

$\ds $

$=$

$\ds 8 \cosh^4 x - 8 \cosh^2 x + 1$

multiplying out and gathering terms

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.34$: Multiple Angle Formulas