Example 3 (2004 National High School Competition Question) Let the acute angle $\theta$ make the equation about $x$, $x^{2}+4 x \cdot \cos \theta+\cot \theta=$ 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. $\frac{\pi}{12}$
Solution: Choose B. Reason: From the given equation, it has a repeated root, so its discriminant $\Delta=16 \cos ^{2} \theta-4 \cot \theta=0$. Since $0<\theta<\frac{\pi}{2}$, we have $4 \cos \theta(2 \sin 2 \theta-1)=0$, which gives $\sin 2 \theta=\frac{1}{2}$. Therefore, $\theta=\frac{\pi}{12}$ or $\frac{5 \pi}{12}$.