Question

Which of the followings is NOT a solution to the trigonometric equation cos4θ+cos2θ=cosθ ?
(A) –π/9 (B) π/9 (C) 3π/9 (D) 5π/9 (E) 7π/9 (hint: try sum-to-product)

Answer 1

cos4θ+cos2θ=cosθ , I will replace θ with x for easier typing

2cos[(4x+2x)/2]cos[(4x-2x)/2] = cosx , using their hint
2cos(3x)cos(x) = cosx
2cos(3x)cosx - cosx = 0
cosx(2cos 3x - 1)
cosx = 0 or cos 3x = 1/2

if cosx = 0, x = π/2 , 3π/2
if cos 3x = 1/2,
3x = ±π/3, ±5π/3, ±7π/3, ±11π/3,...
x = ± π/9, ± 5π/9, ± 7π/9, ± 11π/9

looking at our list, 3π/9 is not a solution, so that's it.

I might have been just as easy to just test the given solutions
to see if they work.