4. The following are 4 propositions:
(1) Given $45^{\circ}<\alpha<90^{\circ}, \sin \alpha \cdot \cos \alpha=\frac{1}{4}$, then $\cos \alpha-\sin \alpha=\frac{\sqrt{2}}{2}$;
(2) Given that the diagonal $A C$ of quadrilateral $A B C D$ bisects $\angle A$ and $\angle C$, and $\angle A=\angle C$, then this quadrilateral is a rhombus;
(3) In the cyclic quadrilateral $A B C D$, $A B / / C D$, $P$ is a point on the circle different from $A, B, C, D$, then $\angle A P D = \angle B P C$;
(4) If $a^{2}-5 a+5=0$, then $\sqrt{(1-a)^{2}}=1-a$. Among these, the number of incorrect propositions is ( ) .
(A) 1
(B) 2
(C) 3
(D) 4
4.C.

Since $\cos \alpha - \sin \alpha = -\frac{\sqrt{2}}{2}$, (1) is incorrect.
(2) is correct.

If point $P$ is on the arc $AD$ or arc $BC$ between $AB$ and $CD$, then $\angle APD + \angle BPC = 180^{\circ}$; if point $P$ is on arc $AB$ or arc $CD$, then $\angle APD = \angle BPC$. Hence (3) is incorrect.
Since $1 - a = -\frac{a^2}{5} \leq 0$, then $\sqrt{(1 - a)^2} = a - 1$. Hence (4) is incorrect.