Question

Cos(8x)-cos(9x)=sin(x)

Answer 1

cos 8x - cos 9x = sin x

-2 · sin 17x/2 · sin (-x/2) = 2 · sin x/2 · cos x/2

sin 17x/2 · sin x/2 = sin x/2 · cos x/2

sin x/2 · (sin 17x/2 - cos x/2) = 0

sin x/2 · (sin 17x/2 - sin (π/2 - x/2)) = 0

sin x/2 · (2 · sin (9x/2 - π/4) · cos (4x + π/4)) = 0

sin x/2 · sin (9x/2 - π/4) · cos (4x + π/4) = 0

sin x/2 = 0

sin (9x/2 - π/4) = 0

cos (4x + π/4) = 0

x/2 = πn, n ∈ Z

9x/2 - π/4 = πk, k ∈ Z

4x + π/4 = π/2 + πm, m ∈ Z

x = 2πn, n ∈ Z

9x/2 = π/4 + πk, k ∈ Z

4x = π/4 + πm, m ∈ Z

x = 2πn, n ∈ Z

x = π/18 + 2πk/9, k ∈ Z

x = π/16 + πm/4, m ∈ Z