Problem

Figure 1 shows a sketch of the curve $C$ with equation

$$y=\frac12\tan x+\cot x \qquad \frac{\pi}{6}\le x\le \frac{\pi}{3}$$

(a) Show that the length of $C$ is given by

$$\frac12\int_{\pi/6}^{\pi/3}\left(\tan^2x+\cot^2x\right)\,dx$$

(b) Hence determine the exact length of $C$, giving your answer in simplest form.