Quadruple Angle Formulas/Hyperbolic Cosine
Jump to navigation
Jump to search
Theorem
- $\cosh 4 x = 8 \cosh^4 x - 8 \cosh^2 x + 1$
where $\cosh$ denotes hyperbolic cosine.
Proof
\(\ds \cosh 4 x\) | \(=\) | \(\ds \map \cosh {2 x + 2 x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh 2 x \cosh 2 x + \sinh 2 x \sinh 2 x\) | Hyperbolic Cosine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\cosh^2 x + \sinh^2 x} \paren {\cosh^2 x + \sinh^2 x} + \paren {2 \sinh x \cosh x} \paren {2 \sinh x \cosh x}\) | Double Angle Formulas | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh^4 x + 2 \cosh^2 x \sinh^2 x + \sinh^4 x + 4 \cosh^2 x \sinh^2 x\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh^4 x + 2 \cosh^2 x \paren {\cosh^2 x - 1} + \paren {\cosh^2 x - 1}^2 + 4 \cosh^2 x \paren {\cosh^2 x - 1}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 8 \cosh^4 x - 8 \cosh^2 x + 1\) | multiplying out and gathering terms |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.34$: Multiple Angle Formulas