Condense the right side a single sine expression:
sin(6x) - cos(6x) = R sin(6x - t)
Expanding the right side gives
sin(6x) - cos(6x) = R sin(6x) cos(t) - R cos(6x) sin(t)
Then we have
R cos(t) = 1
R sin(t) = 1
Solve for R and t:
(R cos(t))² + (R sin(t))² = 1² + 1²
R² = 2
R = √2
and
(R sin(t))/(R cos(t)) = 1/1
tan(t) = 1
t = arctan(1) = π/4
So we rewrite the equation as
√2 sin(6x - π/4) = √2
Solve for x :
sin(6x - π/4) = 1
6x - π/4 = arcsin(1) + 2nπ
(where n is any integer)
6x - π/4 = π/2 + 2nπ
6x = 3π/4 + 2nπ
x = π/8 + nπ/3