If $x\cos \theta =y\cos \left(\theta +\frac{2\pi }{3}\right)=z\cos \left(\theta +\frac{4\pi }{3}\right)$ then the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=$

$x\cos \theta =y\cos \left(\theta +\frac{{\displaystyle 2\pi }}{{\displaystyle 3}}\right)=z\cos \left(\theta +\frac{{\displaystyle 4\pi }}{{\displaystyle 3}}\right)=k$   $\frac{k}{x}=\cos \theta ,\frac{k}{y}=\cos \left(\theta +\frac{{\displaystyle 2\pi }}{{\displaystyle 3}}\right),\frac{k}{z}=\cos \left(\theta +\frac{{\displaystyle 4\pi }}{{\displaystyle 3}}\right)$

$$\begin{gathered} \therefore k\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\cos \theta +\cos \left(\frac{2\pi }{3}+\theta \right)+\cos \left(\frac{4\pi }{3}+\theta \right)=\cos \theta +\cos (120°+\theta )+\cos (120°-\theta ) \\ =\cos \theta +2\cos 120\cos \theta =\cos \theta +2(-1/2)\cos \theta =0 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0 \end{gathered}$$