Range of $\frac{1}{2\cos x-1}$ How can we find the range of $$f(x) =\frac{1}{2\cos x-1}$$
Since range of $\cos x$ can be given as : $-1 \leq \cos x \leq 1$ 
therefore we can proceed as :$$\begin{array}{rcl}
 -2 \leq      & 2\cos x                  & \leq 2 \\
 -2-1 \leq    & 2\cos x -1               & \leq 2-1\\ 
 -3 \leq      & 2\cos x -1               & \leq 1 \\
 \frac{-1}{3} \leq & \frac{1}{2\cos x-1} & \leq 1
\end{array}$$
Is it the range? Please suggest and guide. 

As $2\cos x-1$ varies through $[-3,0)$, $\frac{1}{2\cos x-1}$ varies through $(-\infty,-\frac13]$.
As $2\cos x-1$ varies through $(0,1]$, $\frac{1}{2\cos x-1}$ varies through $[1,+\infty)$.
Note that division by $0$ is undefined, so we do not consider the case $2\cos x-1=0$.
Hence the range is $(-\infty,-\frac13]\cup [1,+\infty)$.
