$\begin{aligned} & \lim _{x \rightarrow 1}(1-x) \frac{\sin \frac{\pi x}{2}}{\cos \frac{\pi x}{2}}=\lim _{1-x \rightarrow 0}(1-x) \frac{\sin \frac{\pi x}{2}}{\cos \frac{\pi x}{2}} \\ = & \lim _{1-x \rightarrow 0}(1-x) \frac{\sin \frac{\pi x}{2}}{\sin \left(\frac{\pi}{2}-\frac{\pi x}{2}\right)} \\ = & \lim _{1-x \rightarrow 0}(1-x) \times \frac{1}{\sin \frac{\pi(1-x)}{2}}\left(\sin \frac{\pi x}{2}\right) \\ = & \lim _{1-x \rightarrow 0} \frac{\pi(1-x)}{2} \times \frac{1}{\sin \frac{\pi}{2}(1-x)} \frac{2}{\pi}\left(\sin \frac{\pi x}{2}\right) \\ = & 1 \times \frac{2}{\pi} \times \sin \frac{\pi}{2}=\frac{2}{\pi} .\end{aligned}$