#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^@}#
