The value of $\tan 20^{\circ} \tan 80^{\circ} \cot 50^{\circ}=$

Evaluate the expression $\tan 20^{\circ} \tan 80^{\circ} \cot 50^{\circ}$.

Note that $\cot 50^{\circ} = \tan 40^{\circ}$, converting via the complementary identity.
Substituting gives $\tan 20^{\circ} \tan 80^{\circ} \tan 40^{\circ}$.

Rearranged, this is $\tan 20^{\circ} \tan 40^{\circ} \tan 80^{\circ}$.

This matches the identity $\tan \theta \tan (60^{\circ} - \theta) \tan (60^{\circ} + \theta) = \tan 3\theta$ with $\theta = 20^{\circ}$,
since $60^{\circ} - 20^{\circ} = 40^{\circ}$ and $60^{\circ} + 20^{\circ} = 80^{\circ}$.

Thus, $\tan 20^{\circ} \tan 40^{\circ} \tan 80^{\circ} = \tan 60^{\circ} = \sqrt{3}$.

The value is $\sqrt{3}$, so option A is correct.