My friend needs help with this question and I don't know it.
How to express $\cosh(4x)$ as a polynomial in $\cosh(x)$
3 Answers
For a mechanical way to translate the more familiar trigonometric identities to hyperbolic function identities, use $\cosh x=\cos(ix),\qquad \sinh x=-i\sin(ix).$
But perhaps an identity for $\cos(4w)$ does not qualify as familiar. However, it is easy to derive from standard facts: $\cos(4w)=2\cos^2 (2w) -1=2(2\cos^2 w -1)^2-1.$ Putting $w=ix$, we obtain in a mechanical way $\cosh(4x)=2(2\cosh^2 x -1)^2-1.$
$\cosh(2x)=2\cosh^2(x)-1$
$\cosh(4x)=2\cosh^2(2x)-1=2(2\cosh^2(x)-1)^2-1=8\cosh^4(x)-8\cosh^2(x)+1$
If you use the definition of the hyperbolic cosine then, dropping the factor 2, you have:
$\exp(4x)+\exp(-4x) = (\exp(2x)-\exp(-2x))^2+2$
then,
$\exp(2x)-\exp(-2x) = (\exp(x)+\exp(-x))(\exp(x)-\exp(-x))$
if you do it you should get something like (if I'm not mistaken)
$\cosh(4x) = 8(\cosh(x))^2(\sinh(x))^2+1$
then, if you want to replace the $\sinh$ you can use the fact that $(\cosh(x))^2-(\sinh(x))^2=1$.