Answer
$1$
Work Step by Step
$\because \cot{\theta} = \dfrac{1}{\tan{\theta}}$
$\therefore \cot{10^{\circ}} = \dfrac{1}{\tan{10^{\circ}}}$
Hence,
$\tan{10^{\circ}} \cdot \cot{10^{\circ}} = \tan{10^{\circ}} \cdot \dfrac{1}{\tan{10^{\circ}}} = \boxed{1}$
