<s>[INST] 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. 
 [/INST] No, you are fine down to $-3 \le 2\cos x -1 \le 1$, but you need to change the sense of the inequality when  you invert:  Think of a positive $x \le 1$, then $\frac 1x \ge 1$.  The idea is that if the denominator is small (and you have trapped it near $0$) the inverse is large.
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