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

$\blacksquare$


Also see


Sources