Question

Show that cos2x=cosx

Answer 1

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