Let $k$ be a non-negative integer. We define the function $f _ { k }$ from the interval $[ - 1,1 ]$ to itself by $f _ { k } ( x ) = \cos ( k \arccos x )$, and $T_k$ is the polynomial of degree $k$ that identifies with $f_k$ on $[-1,1]$. We denote by arcosh the inverse function of the hyperbolic cosine, defined from $[ 1 , + \infty [$ to $[ 0 , + \infty [$.
Show that
$$\forall x \in ] - \infty , - 1 ] \quad T _ { k } ( x ) = ( - 1 ) ^ { k } \cosh ( k \operatorname { arcosh } ( - x ) ) .$$