## 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

$\blacksquare$