If $32 {\tan }^8\mathrm{\theta }=2 {\cos }^2\mathrm{\alpha }-3 \mathrm{cos\alpha }$ and $3\cos 2\mathrm{\theta }=1,$ then the general value of $\mathrm{\alpha }$ is

$$\begin{gathered} 32{\tan }^8\mathrm{\theta }=2{\cos }^2\mathrm{\alpha }-3\cos \mathrm{\alpha } \\ 3\cos 2\mathrm{\theta }=1 \\ 3\left(\frac{1-{\tan }^2\mathrm{\theta }}{1+{\tan }^2\mathrm{\theta }}\right)=1 \\ 3-3{\tan }^2\mathrm{\theta }=1+{\tan }^2\mathrm{\theta } \\ 4{\tan }^2\mathrm{\theta }=2 \Rightarrow {\tan }^2\mathrm{\theta }=\frac{1}{2} \end{gathered}$$

$$\begin{gathered} 32\left(\frac{1}{16}\right)=2{\cos }^2\mathrm{\alpha }-3\cos \mathrm{\alpha } \\ 2={\cos }^2\mathrm{\alpha }-3\cos \mathrm{\alpha } \\ 2{\cos }^2\mathrm{\alpha }-3\cos \mathrm{\alpha }-2=0 \\ (2\cos \mathrm{\alpha }-1) (\cos \mathrm{\alpha }-1)=0 \\ \cos \mathrm{\alpha }=\frac{1}{2} \cos \mathrm{\alpha }=1 \\ \mathrm{\alpha }=2\mathrm{n\pi }\pm \frac{\mathrm{\pi }}{3} \end{gathered}$$