Given the conditions, we can let $a=\cot \frac{A}{2}, b=\cot \frac{B}{2}, c=\cot \frac{C}{2}$ and satisfy $A+B+C=\pi$. Using the following two identities:
$$\begin{array}{c} 1+\cot ^{2} x=\csc ^{2} x \\ \sin \frac{A}{2}+\sin \frac{B}{2}+\sin \frac{C}{2}=4 \sin \frac{A+B}{4} \sin \frac{B+C}{4} \sin \frac{C+A}{4}+1 \end{array}$$

The inequality can be simplified to:
$$\begin{array}{l} \csc \frac{A}{2} \csc \frac{B}{2}+\csc \frac{B}{2} \csc \frac{C}{2}+\csc \frac{C}{2} \csc \frac{A}{2} \geqslant 4+\csc \frac{A}{2} \csc \frac{B}{2} \csc \frac{C}{2} \Leftrightarrow \\ \sin \frac{A}{2}+\sin \frac{B}{2}+\sin \frac{C}{2} \geqslant 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}+1 \Leftrightarrow \\ \sin \frac{A+B}{2} \sin \frac{B+C}{4} \sin \frac{C+A}{4} \geqslant \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \end{array}$$

Using the AM - GM inequality, we have:
$$\begin{array}{l} \sin \frac{A+B}{4}=\sin \frac{A}{4} \cos \frac{B}{4}+\cos \frac{A}{4} \sin \frac{B}{4} \geqslant \\ \sqrt{4 \sin \frac{A}{4} \cos \frac{A}{4} \sin \frac{B}{4} \cos \frac{B}{4}}=\sqrt{\sin \frac{A}{2} \sin \frac{B}{2}} \end{array}$$

Similarly, we can obtain:
$$\begin{array}{l} \sin \frac{B+C}{4} \geqslant \sqrt{\sin \frac{B}{2} \sin \frac{C}{2}} \\ \sin \frac{C+A}{4} \geqslant \sqrt{\sin \frac{C}{2} \sin \frac{A}{2}} \end{array}$$

Multiplying the above three inequalities will yield the desired inequality. Familiarity with the relationships between trigonometric functions is also essential.