$$
\begin{aligned}
& \text { Given, } x \cos \theta=y \cos \left(\theta+\frac{2 \pi}{3}\right) \\
& z \cos \left(\theta+\frac{4 \pi}{3}\right)=\lambda \text { (say) } \\
& \Rightarrow \cos \theta=\frac{\lambda}{x}, \cos \left(\theta+\frac{2 \pi}{3}\right)=\frac{\lambda}{y}, \cos \left(\theta+\frac{4 \pi}{3}\right)=\frac{\lambda}{z} \\
& \therefore \cos \theta+\cos \left(\theta+\frac{2 \pi}{3}\right)+\cos \left(\theta+\frac{4 \pi}{3}\right) \\
& =\lambda\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \\
& \Rightarrow \lambda\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=2 \cos \left(\frac{\theta+\theta+4 \pi / 3}{2}\right) \\
& \cos \left(\frac{\theta+4 \pi / 3-\theta}{2}\right)+\cos \left(\theta+\frac{2 \pi}{3}\right) \\
& \Rightarrow \quad\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right] \\
& \Rightarrow \quad \lambda\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=2 \cos \left(\theta+\frac{2 \pi}{3}\right) \\
& \Rightarrow \quad \cos \left(\frac{2 \pi}{3}\right)+\cos \left(\theta+\frac{2 \pi}{3}\right)
\end{aligned}
$$

$$
\begin{aligned}
& \because \cos \left(\frac{2 \pi}{3}\right)=\cos \left(\pi-\frac{\pi}{3}\right)=-\cos \frac{\pi}{3}=-\frac{1}{2} \\
& \therefore \lambda\left(\frac{1}{x}+\frac{1}{y}+\frac{+1}{z}\right)=-\cos \left(\theta+\frac{2 \pi}{3}\right)+\cos \left(\theta+\frac{2 \pi}{3}\right) \\
& =0 \\
& \therefore \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0 \quad \quad[\because \lambda \neq 0]
\end{aligned}
$$