Segundo enunciado,\begin{matrix} z&=&[(1+\cos{2x}) + i\sin{2x}]^k &=& [2\cos^2{x}+ i(2\sin{x}\cos{x})]^k &=& (2\cos{x})^k(\cos{x} + i\sin{x})^k \end{matrix} Com conhecimento da $\text{Lei de Moivre}$, têm-se:\begin{matrix}(2\cos{x})^k(\cos{kx} + i\sin{kx}) &\Rightarrow& \fbox{$Im(z) = 2^k\sin{kx}\cos^k{x}$} \end{matrix} \begin{matrix} Letra \ (C) \end{matrix} $\color{orangered}{Obs:}$ ${\begin{cases} \cos{2x} &=& 2cos^2{x} -1 &,& \sin{2x} &=& 2\sin{x}\cos{x} \end{cases}}$