The general solution of the equation $\cos x\cos 6x=-1$, is

We have,

$\begin{array}{l}\cos x\cos 6x=-1\\ \Rightarrow 2\cos x\cos 6x=-2\\ \Rightarrow \cos 7x+\cos 5x=-2\Rightarrow \cos 7x=-1\end{array}$

and  $\cos 5 x=-1$ 

The value of x satisfying these two equations simultaneously and lying between 0 and 2 $\mathrm{\pi }$ is $\mathrm{\pi }$. Therefore, the general solution is given by

$x=2n\pi +\pi ,n\in Z\Rightarrow x=(2n+1)\pi ,n\in Z$