1 Let the acute angle $\theta$ make the equation $x^{2}+4 x \cos \theta+\cot \theta=0$ have a repeated root, then the radian measure of $\theta$ is ( ).
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{12}$ or $\frac{5 \pi}{12}$
(C) $\frac{\pi}{6}$ or $\frac{5 \pi}{12}$
(D) $\frac{\pi}{12}$
Given that the equation $x^{2}+4 x \cos \theta+\cot \theta=0$ has a repeated root, hence
$$
\Delta=16 \cos ^{2} \theta-4 \cot \theta=0,
$$

which means
$$
4 \cot \theta(2 \sin 2 \theta-1)=0.
$$

Since $0<\theta<\frac{\pi}{2}$, we get
$$
\sin 2 \theta=\frac{1}{2},
$$

so $2 \theta=\frac{\pi}{6}$ or $2 \theta=\frac{5 \pi}{6}$,

thus $\theta=\frac{\pi}{12}$ or $\theta=\frac{5 \pi}{12}$. Therefore, the answer is B.