Quadruple Angle Formulas/Hyperbolic Cosine

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Theorem

$\cosh \left({4 x}\right) = 8 \cosh^4 x - 8 \cosh^2 x + 1$

where $\cosh$ denotes hyperbolic cosine.


Proof

\(\displaystyle \cosh \left({4 x}\right)\) \(=\) \(\displaystyle \cosh \left({2 x + 2 x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh 2 x \cosh 2 x + \sinh 2 x \sinh 2 x\) Hyperbolic Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \left({\cosh^2 x + \sinh^2 x}\right) \left({\cosh^2 x + \sinh^2 x}\right) + \left({2 \sinh x \cosh x}\right) \left({2 \sinh x \cosh x}\right)\) Double Angle Formulas
\(\displaystyle \) \(=\) \(\displaystyle \cosh^4 x + 2 \cosh^2 x \sinh^2 x + \sinh^4 x + 4 \cosh^2 x \sinh^2 x\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle \cosh^4 x + 2 \cosh^2 x \left({\cosh^2 x - 1}\right) + \left({\cosh^2 x - 1}\right)^2 + 4 \cosh^2 x \left({\cosh^2 x - 1}\right)\) Difference of Squares of Hyperbolic Cosine and Sine
\(\displaystyle \) \(=\) \(\displaystyle 8 \cosh^4 x - 8 \cosh^2 x + 1\) multiplying out and gathering terms

$\blacksquare$


Also see


Sources