The general value of $\theta$ which satisfies the equation $\left(\cos \mathrm{\theta }+\mathrm{isin} \mathrm{\theta }\right)\left(\cos 3\mathrm{\theta }+\mathrm{i} \sin 3\mathrm{\theta }\right).....$

$\left[\left(\cos \left(2\mathrm{r}-1\right)\mathrm{\theta }+\mathrm{isin}\left(2\mathrm{r}-1\right)\right)\mathrm{\theta }\right]=1 \mathrm{is}$

$\begin{array}{rcl}\mathrm{cis\theta } \mathrm{cis}3\mathrm{\theta } ..... \mathrm{cis} (2\mathrm{r}-1)\mathrm{\theta }& =& 1\\ \mathrm{cis}(\mathrm{\theta }+3\mathrm{\theta }+....+(2\mathrm{r}-1\left)\mathrm{\theta }\right)& =& 1\\ \mathrm{cis}(1+3+5+....+(2\mathrm{r}-1\left)\right)\mathrm{\theta }& =& 1\\ \mathrm{cis}\left(\frac{\mathrm{r}}{2}(1+2\mathrm{r}-1)\mathrm{\theta }\right)& =& 1\\ \mathrm{cis} {\mathrm{r}}^2\mathrm{\theta }& =& 1\\ & \Rightarrow & \cos {\mathrm{r}}^2\mathrm{\theta }=1, \sin {\mathrm{r}}^2\mathrm{\theta }=0\\ \therefore {\mathrm{r}}^2\mathrm{\theta }& =& 2\mathrm{n\pi }\\ & \Rightarrow & \mathrm{\theta }=\frac{2\mathrm{n\pi }}{{\mathrm{r}}^2}\end{array}$