3. $f(x)$ is a function defined on $\mathbf{R}$ that is not identically zero, $f\left(\frac{\pi}{2}\right)=0$, and for any $x, y \in \mathbf{R}$, we have $f(x)+$ $f(y)=2 f\left(\frac{x+y}{2}\right) \cdot f\left(\frac{x-y}{2}\right)$, then among the following expressions, the one that does not hold is
A. $f(x+2 \pi)=f(x)$
B. $f(-x)=f(x)$
C. $f(2 x)=2 f^{2}(x)-1$
D. $f(0)=0$
3. $\mathrm{D}$ Hint: $f(0)=1$