Solution:

On $[0, 2\pi],$ the graph of $\cos\left(\tfrac{x}{2}\right)$ is a single arc decreasing from $1$ down to $-1.$

The function $\tan(2x)$ has period $\tfrac{\pi}{2}$ with vertical asymptotes at $x = \tfrac{\pi}{4}, \tfrac{3\pi}{4}, \tfrac{5\pi}{4}, \tfrac{7\pi}{4}.$ These split the interval into five stretches, and on each stretch $\tan(2x)$ runs through all real values.

Since the cosine arc is bounded, each of the five branches meets it exactly once, giving $5$ solutions.

Thus, E is the correct answer.