$1+\cos 2\mathrm{\theta }+\cos 4\mathrm{\theta }+\cos 6\mathrm{\theta }=$

$$\begin{gathered} =2{\cos }^2\mathrm{\theta }+2\cos \left(\frac{4\mathrm{\theta }+6\mathrm{\theta }}{2}\right)\cos \left(\frac{4\mathrm{\theta }-6\mathrm{\theta }}{2}\right)=2{\cos }^2\mathrm{\theta }+2\cos 5\mathrm{\theta cos}\mathrm{\theta } \\ =2\cos \mathrm{\theta }(\cos \mathrm{\theta }+\cos 5\mathrm{\theta })=2\cos \mathrm{\theta }\cdot 2\cos \left(\frac{\mathrm{\theta }+5\mathrm{\theta }}{2}\right)\cos \left(\frac{\mathrm{\theta }-5\mathrm{\theta }}{2}\right)=4\cos \mathrm{\theta cos}2\mathrm{\theta cos}3\mathrm{\theta } \end{gathered}$$