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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!

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    $\begingroup$ See the "Hyperbolic identities" section of Wikipedia's "Tangent half-angle formula" entry. $\endgroup$
    – Blue
    Commented Jan 18, 2022 at 23:47
  • $\begingroup$ @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 $\sinh (2 \arctanh (t)) = -((2 t)/(-1 + t^2))$ $\endgroup$
    – juan zaragoza
    Commented Jan 19, 2022 at 0:05

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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.

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