4. For all positive integers $n, \tan { }^{n} 15^{\circ}+\cot ^{n} 15^{\circ}$ is always
4. Positive even number Detailed explanation: $\tan ^{n} 15^{\circ}+\cot ^{n} 15^{\circ}=(2+\sqrt{3})^{n}+(2-\sqrt{3})^{n}=\sum_{r=0}^{n} C_{n}^{r} 2^{n-r}\left[(\sqrt{3})^{r}+(-\sqrt{3})^{r}\right]=$ $2 \sum_{2 \mid r} C_{n}^{r} 2^{n-1} 3^{\frac{r}{2}}$ is a positive even number. Another solution: using the recurrence relation of the sequence $a_{n}=\tan ^{n} 15^{\circ}+\cot ^{n} 15^{\circ}$, $a_{n}=4 a_{n-1}-a_{n-2}(n>2)$, and the initial conditions $a_{0}=2, a_{1}=4$, it can be deduced by recursion.