Cos x (2 sent x+1) =0

To solve the equation \cos x (2\sin x+1) = 0, we need to set each factor equal to zero and solve for x.

First, set \cos x = 0:
\cos x = 0
x = \frac{\pi}{2} + n\pi where n is an integer.

Next, set 2\sin x + 1 = 0:
2\sin x + 1 = 0
2\sin x = -1
\sin x = -\frac{1}{2}

The solutions for \sin x = -\frac{1}{2} are:
x = \frac{7\pi}{6} + 2n\pi and
x = \frac{11\pi}{6} + 2n\pi
where n is an integer.

Therefore, the solutions to the equation \cos x (2\sin x+1) = 0 are:
x = \frac{\pi}{2} + n\pi, \frac{7\pi}{6} + 2n\pi, \frac{11\pi}{6} + 2n\pi
where n is an integer. \boxed{x = \frac{\pi}{2} + n\pi, \frac{7\pi}{6} + 2n\pi, \frac{11\pi}{6} + 2n\pi}