Answer
$1$
Work Step by Step
I know that $\tan$ has a period of $180^\circ$, hence $\tan{\theta}=\tan{(\theta-180^\circ)}.$
Hence: $\tan{200^\circ}\cot{20^\circ}=\tan{(200^\circ-180^\circ)}\cot{20^\circ}=\tan{20^\circ}\cot{20^\circ}$.
We also know that $\tan{\theta}\cot{\theta}=1$.
Thus: $\tan{20^\circ}\cot{20^\circ}=1$.
