min value of

$\begin{aligned} y &= \cos x \sin x + \sin x + \cos x + 4 \\ \\ &= (\cos x + 1)(\sin x + 1) + 3 \end{aligned}$

Since neither factor above can be negative, the minimum value of their product will be zero, when $\cos x = -1$ or $\sin x = -1$ , i.e. when $x = \pi + 2n \pi$ or $x = \dfrac{3 \pi}{2} + 2n \pi$ .

Thus the minimum value of $y$ is $\boxed{3}$