# Hyperbolic functions as polynomials

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!

• See the "Hyperbolic identities" section of Wikipedia's "Tangent half-angle formula" entry.
– Blue
Commented Jan 18, 2022 at 23:47
• @Blue yeah, it has been really helpful. To anyone interested that doesn't want to read any further, $\cosh (2 \arctanh (t) )= (1 + t^2)/(1 - t^2)$ and $\sinh (2 \arctanh (t)) = -((2 t)/(-1 + t^2))$ Commented Jan 19, 2022 at 0:05

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.