cos(x)(csc(x)+2) = 0
=> cos(x) = 0 or csc(x) + 2 = 0
=> cos(x) = 0 or csc(x) = -2
Case 1: cos(x) = 0
Noting that the cosine function takes on the x value of the coordinate reached by rotating by a given angle around the unit circle, it is clear that it takes on a value of 0 only when the angle lands on the y-axis, that is, at 90^@ or 270^@.
Case 2: csc(x) = -2
Using the definition of csc(x) = 1/sin(x), we have
1/sin(x) = -2
=> sin(x) = -1/2
Noting that the sine function takes on the y value of the point reached by rotating by the given angle around the unit circle, we check our unit circle again and find that sin(x) = -1/2 when x=210^@ or x=330^@
As we are restricted to the interval [0^@, 360^@), we do not need to consider adding or subtracting 360^@ from any of the angle. Thus, our entire solution set is
x in {90^@, 210^@, 270^@, 330^@}
