Number of values of $\theta$ for the equation $\cos \theta +\cos 3\theta +\cos 5\theta +\cos 7\theta =0$ in $\left(0,\pi \right)$ is

$We have, \cos \theta +\cos 3\theta +\cos 5\theta +\cos 7\theta =0$

$\Rightarrow \left(\cos \theta +\cos 7\theta \right)+\left(\cos 3\theta +\cos 5\theta \right)=0$

$\Rightarrow 2\cos 4\theta \cos 3\theta +2\cos 4\theta .\cos \theta =0$

$\Rightarrow \cos 4\theta \left(\cos 3\theta +\cos \theta \right)=0$

$\Rightarrow \cos 4\theta \left(2\cos 2\theta \cos \theta \right)=0$

Either $\cos \theta =0$$\Rightarrow \theta =\left(2n+1\right)\frac{\pi }{2}$

   $or \cos 2\theta =0\Rightarrow \theta =\left(2n+1\right)\frac{\pi }{4}$

$or \cos 4\theta =0\Rightarrow \theta =\left(2n+1\right)\frac{\pi }{8}$

$Since \theta \in \left(0,\mathrm{\pi }\right) \Rightarrow \theta =\frac{\mathrm{\pi }}{8},\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{8},\frac{\mathrm{\pi }}{2},\frac{5\mathrm{\pi }}{8},\frac{3\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{8}$