Let $k$ be a non-negative integer. We define the function $f _ { k }$ from the interval $[ - 1,1 ]$ to itself by $f _ { k } ( x ) = \cos ( k \arccos x )$.
a) Expand the expression $f _ { k + 1 } ( x ) + f _ { k - 1 } ( x )$, and deduce the relation
$$\forall x \in [ - 1,1 ] \quad f _ { k + 1 } ( x ) = 2 x f _ { k } ( x ) - f _ { k - 1 } ( x )$$
b) Deduce that $f _ { k }$ identifies on $[ - 1,1 ]$ with a polynomial $T _ { k }$, of degree $k$, with the same parity as $k$.