Find all exact solutions on the interval [0, 2π). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) cos(2x) + cos(x) = 0

Answer 1

cos2x+cosx=0

cos2x=2cos^2x-1 (trig identity)

substitute

2cos^2x-1+cosx=0

2cos^2x+cosx-1=0

factor

(2cosx-1)(cosx+1)=0

2cosx-1=0

2cosx=1

cosx=1/2

cos is equal to 1/2 at ∏/3 and 5∏/3

cosx+1=0

cosx=-1

cos is equal to -1 at ∏