Now,
$$
\begin{aligned}
& \cos (\theta-\alpha)+\cos (\theta-\beta)+\cos \theta+\cos (\theta-\gamma) \\
&= 2 \cos \left(\theta-\left(\frac{\alpha+\beta}{2}\right)\right) \cos \left(\frac{\beta-\alpha}{2}\right) \\
&+2 \cos \left(\frac{\gamma}{2}\right) \cos \left(\theta-\frac{\gamma}{2}\right) \\
&=2 \cos \left(\frac{\gamma}{2}\right) \cos \left(\frac{\beta-\alpha}{2}\right) \\
&+2 \cos \left(\frac{\gamma}{2}\right) \cos \left(\frac{\alpha+\beta}{2}\right) \\
&=2 \cos \left(\frac{\gamma}{2}\right)\left[\cos \left(\frac{\beta-\alpha}{2}\right)+\cos \left(\frac{\alpha+\beta}{2}\right)\right] \\
&=2 \cos \left(\frac{\gamma}{2}\right) 2 \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \\
&= 4 \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \cos \frac{\gamma}{2}
\end{aligned}
$$