Example 3. Solve the equation: $\operatorname{ctg}^{2} \frac{x}{2}=1+\sec x$.
Correct solution: Upon inspection, $x=2 n \pi+\pi(n \in Z)$ satisfies the original equation, so the solutions to the original equation are $\mathrm{x}=2 \mathrm{n} \pi \pm \frac{\pi}{3}, x=2 n \pi+\pi,(n \in Z)$