I have recently found that the change of variable $t\to 2 \arctan (t)$ makes $\cos(2 \arctan (t)) = \dfrac{1-t^2}{1+t^2}$ and $\sin (2\arctan (t) ) = \dfrac{2t}{1+t^2}$ for certain values of $t$. I was wondering, is there any change of variables that transforms $\cosh(t)$ and $\sinh(t)$ into polynomials or a quotient of polynomials locally? I have done some research but all I have found is about hyperbolic polynomials and I don't see how that would help me.
Thanks everyone!
As it has been said in the comments, $\cosh(2\tanh^{-1} (t))=(1+t^2)/(1−t^2)$ and $\sinh(2\tanh^{-1}(t))=((2t)/(1-t^2))$ hold.