Show that cos2x=cosx

cos (2x) = cos x

2 cos^2 x - 1 = cos x using the double angle formula

2 cos ^2 x - cos x - 1 = 0

factor

(2 cos x+1) (cos x - 1) = 0

using the zero product property

2 cos x+1 = 0 cos x - 1 = 0

2 cos x = - 1 cos x = 1

cos x = - 1/2 cos x=1

taking the arccos of each side

arccos cos x = arccos (-1/2) arccos cos x = arccos 1

x = 120 degrees x=-120 degrees x=0

remember you get 2 values (2nd and 3rd quadrant)

these are the principal values

now we need to add 360

x = 120 + 360n x=-120 + 360n x = 0 + 360n where n is an integer