11. Given $\sin 2 \alpha=2 \sin 4^{\circ}, \tan \left(\alpha+2^{\circ}\right) \cot \left(\alpha-2^{\circ}\right)=$
11. 3.

Since $\sin 2 \alpha=2 \sin 4^{\circ}$, let $\alpha+2^{\circ}=x, \alpha-2^{\circ}=y$, then $2 \alpha=x+y, 4^{\circ}=x-y$, so we have $\sin (x+y)=2 \sin (x-y)$, which means $\sin x \cos y=3 \cos x \sin y$, thus $\tan x \cdot \cot y=3$. Therefore, $\tan (\alpha+2^{\circ}) \cdot \cot (\alpha-2^{\circ})=3$.