AMC12 2020 A

Let $f(x) = \tan(2x)$ and $g(x) = \cos\left(\frac{x}{2}\right)$, so that the given equation is $f(x) = g(x)$. On the interval $[0, 2\pi]$, the values of $g(x)$ decrease from 1 to -1. On each of the intervals $$\left( \frac{(2k-1)\pi}{4}, \frac{(2k+1)\pi}{4} \right), \quad k = 1, 2, 3,$$ the values of $f(x)$ increase and range over all real numbers, so there is a unique solution in each of those intervals. On the interval $[0, \frac{\pi}{4})$, the values of $f(x)$ increase with range $[0, \infty)$ and $g(x) > 0$, so there is a unique solution in that interval. On the interval $(\frac{7\pi}{4}, 2\pi]$, the values of $f(x)$ increase with range $(-\infty, 0]$ and $g(x) < 0$, so there is a unique solution in that interval. The total number of solutions is 5, as displayed in the following graph.