# Triple Angle Formulas/Hyperbolic Cosine

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## Theorem

$\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$

where $\cosh$ denotes hyperbolic cosine.

## Proof

 $\displaystyle \cosh 3 x$ $=$ $\displaystyle \cosh \paren {2 x + x}$ $\displaystyle$ $=$ $\displaystyle \cosh 2 x \cosh x + \sinh 2 x \sinh x$ Hyperbolic Cosine of Sum $\displaystyle$ $=$ $\displaystyle \paren {\cosh^2 x + \sinh^2 x} \cosh x + \sinh 2 x \sinh x$ Double Angle Formula for Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \paren {\cosh^2 x + \sinh^2 x} \cosh x + \paren {2 \sinh x \cosh x} \sinh x$ Double Angle Formula for Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle \cosh^3 x + \sinh^2 x \cosh x + 2 \sinh^2 x \cosh x$ multiplying out $\displaystyle$ $=$ $\displaystyle \cosh^3 x + \paren {\cosh^2 x - 1} \cosh x + 2 \paren {\cosh^2 x - 1} \cosh x$ Difference of Squares of Hyperbolic Cosine and Sine $\displaystyle$ $=$ $\displaystyle \cosh^3 x + \cosh^3 x - \cosh x + 2 \cosh^3 x - 2 \cosh x$ multiplying out $\displaystyle$ $=$ $\displaystyle 4 \cosh^3 x - 3 \cosh x$ gathering terms

$\blacksquare$