Answer
We fill in the blank with $\tan$.
Work Step by Step
$$\cot\frac{\pi}{3}=A\frac{\pi}{6}\hspace{1cm}(1)$$
We notice that $$\frac{\pi}{3}+\frac{\pi}{6}=\frac{2\pi+\pi}{6}=\frac{3\pi}{6}=\frac{\pi}{2}$$
That means we can rewrite $(1)$ as
$$\cot\frac{\pi}{3}=A(\frac{\pi}{2}-\frac{\pi}{3})$$
Now from the Cofunction Identity:
$$\cot\theta=\tan(\frac{\pi}{2}-\theta)$$
So $$A=\tan$$
In other words, we fill in the blank with $\tan$.
