5. If the two perpendicular tangents of the curve $y=2 \sin \frac{x}{2}$ intersect at point $P$, then the coordinates of point $P$ cannot be ( ).
(A) $(\pi, \pi)$
(B) $(3 \pi,-\pi)$
(C) $(5 \pi,-\pi)$
(D) $(7 \pi,-\pi)$
5. C.

Notice that the slope of the tangent line to the function $y=2 \sin \frac{x}{2}$ at $x=x_{i}$ is $\cos \frac{x_{i}}{2}$.
Therefore, if the tangent lines at $x_{1}$ and $x_{2}$ are perpendicular to each other, then $\cos \frac{x_{1}}{2} \cdot \cos \frac{x_{2}}{2}=-1$.
Assume without loss of generality that $\cos \frac{x_{1}}{2} \leqslant \cos \frac{x_{2}}{2}$.
Thus, $\cos \frac{x_{1}}{2}=-1, \cos \frac{x_{2}}{2}=1$, which means
$$
\frac{x_{1}}{2}=(2 a+1) \pi, \frac{x_{2}}{2}=2 b \pi(a, b \in \mathbf{Z}) \text {. }
$$

Therefore, the intersection point of the two tangent lines is
$$
P\left(x_{0}, y_{0}\right)=\left(\frac{x_{1}+x_{2}}{2}, \frac{x_{1}-x_{2}}{2}\right) \text {. }
$$

It can be seen that both the x-coordinate and y-coordinate of point $P$ are odd multiples of $\pi$, and their difference is an integer multiple of $4 \pi$. Among the four options, only option C does not meet this characteristic.