$$\eqalign{ & \cos \left( {p\arccos x} \right) = \cosh \left( {p\ln \left( {x + \sqrt {{x^2} - 1} } \right)} \right) = \cr & \frac{1}{2}\left( {{{\left( {x + \sqrt {{x^2} - 1} } \right)}^p} + {{\left( {x + \sqrt {{x^2} - 1} } \right)}^{ - p}}} \right) \cr} $$