Example 1. Simplify $\sin x\left(1+\operatorname{tg} 2 \operatorname{tg} \frac{x}{2}\right)$.
Let $\operatorname{tg} \frac{x}{2}=t$, then
$$
\begin{aligned}
& \text { Original expression }=\frac{2 t}{1+t^{2}}\left(1+\frac{2 t \cdot t}{1-t^{2}}\right)=\frac{2 t}{1-t^{2}} \\
= & \operatorname{tg} x .
\end{aligned}
$$