Use the half-angle formula for the tangent function

\tan\dfrac{\alpha }{2} = \pm\sqrt{ \dfrac{1-\cos \alpha }{1+\cos \alpha} }

\tan\dfrac{\pi }{12} = \sqrt{ \dfrac{1-\cos \dfrac{\pi }{6} }{1+\cos\dfrac{\pi }{6}} }

=\sqrt{ \dfrac{1 -\dfrac{\sqrt{3} }{2} }{1+\dfrac{\sqrt{3} }{2} } }

=\sqrt{\dfrac{2-\sqrt{3} }{2+\sqrt{3} } }