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:$$\begin{array}{l}
z^1&=0+z&\\
z^2&=-1+rz\\&=-1+(2\cos\theta)z\\z^3&=-r+(-1+r^2)z\\&=-(2\cos\theta)+[(2\cos\theta)^2-1]z\\&=-(2\cos\theta)+[2\cos2\theta+1]z\\z^4&=-(-1+r^2)+(-2r+r^3)z\\&=-[(2\cos\theta)^2-1]+[(2\cos\theta)^3-2(2\cos\theta)]z\\&=-[2\cos2\theta+1]+[2(\cos\theta+\cos3\theta)]z\\z^5&=-(-2r+r^3)+(-3r^2+1+r^4)z\\&=-[(2\cos\theta)^3-2(2\cos\theta)]+[(2\cos\theta)^4-3(2\cos\theta)^2+1]z\\&=-[2(\cos\theta+\cos3\theta)]+[2(\cos2\theta+\cos4\theta)+1]z\\
&\quad\quad\quad\vdots\\
z^{2m}
&\displaystyle=-\left[1+2\sum_{k=1}^{m-1}\cos2k\theta\right]+\left[2\sum_{k=0}^{m-1}\cos(2k+1)\theta\right]z\\
z^{2m+1}
&\displaystyle=-\left[2\sum_{k=0}^{m-1}\cos(2k+1)\theta\right]+\left[1+2\sum_{k=1}^{m}\cos2k\theta\right]z\\
\end{array}$$
