Answer
$$2 - \sqrt 3 $$
Work Step by Step
$$\eqalign{
& \tan \frac{\pi }{{12}} \cr
& {\text{Write }}\frac{\pi }{{12}}{\text{ as }}\frac{\pi }{4} - \frac{\pi }{6} \cr
& \tan \frac{\pi }{{12}} = \tan \left( {\frac{\pi }{4} - \frac{\pi }{6}} \right) \cr
& {\text{Use Tangent difference identity}} \cr
& \tan \frac{\pi }{{12}} = \frac{{\tan \left( {\pi /4} \right) - \tan \left( {\pi /6} \right)}}{{1 + \tan \left( {\pi /4} \right)\tan \left( {\pi /6} \right)}} \cr
& {\text{Substitute known values}} \cr
& \tan \frac{\pi }{{12}} = \frac{{1 - \sqrt 3 /3}}{{1 + \sqrt 3 /3}} \cr
& \tan \frac{\pi }{{12}} = \frac{{3 - \sqrt 3 }}{{3 + \sqrt 3 }} \cr
& {\text{Rationalizing}} \cr
& \tan \frac{\pi }{{12}} = \frac{{3 - \sqrt 3 }}{{3 + \sqrt 3 }} \times \frac{{3 - \sqrt 3 }}{{3 - \sqrt 3 }} \cr
& \tan \frac{\pi }{{12}} = \frac{{12 - 6\sqrt 3 }}{6} \cr
& \tan \frac{\pi }{{12}} = 2 - \sqrt 3 \cr} $$