Solve, finding all solutions in [0, 2π) (trig help)?

Question

I have my final next week and I'm going over the study guide and I'm stuck here. Please help!

2 sin x cos x + cos x = 0

Answer 1

2sinxcosx + cosx = 0 in[0 , 2π)

cosx(2sinx + 1) = 0

If ab = 0 then a = 0 or b = 0

cosx = 0 or 2sinx + 1 = 0

if cosx = 0 then x = π / 2 , 3π / 2

if 2sinx + 1 = 0 then x = 7π / 6 , 11π / 6.

There fore x = π / 2 , 7π / 6 , 3π / 2 , 11π / 6 in [0 , 2π).