Example 1 (2004 National High School Competition Question) Let the acute angle $\theta$ make the equation $x^{2}+4 x \cdot \cos \theta+\cot \theta=0$ 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: Since the given equation has a repeated root, we know its discriminant $\Delta=16 \cos ^{2} \theta-4 \cot \theta=0$. Also, $\theta$ is an acute angle, i.e., $0<\theta<\frac{\pi}{2}$. By converting the trigonometric functions, we have $16 \cos ^{2} \theta-4 \cot \theta=4 \cos \theta(2 \sin 2 \theta-1)=0$, and since $\cos \theta \neq 0$, we know $2 \sin 2 \theta-1=0$, which means $\sin 2 \theta=\frac{1}{2}$. Therefore, $\theta=\frac{\pi}{12}$ or $\theta=\frac{5 \pi}{12}$.