First, develop #cos (x - pi/3) = cos x.cos ((pi)/3) + sin ((pi)/3).sin x =#
#= (1/2)cos x + (sqrt3/2) sin x.#
Bring equation (1) to standard form, then, simplify
#sqrt3sinx - (sqrt3sin x)/2 - (1/2)cos x = 0#
#(sqrt3)sin x - cos x = 0# (2).
#sin x - (1/sqrt3)cos x = 0#
Replace #(1/sqrt3) = tan ((pi)/6) = (sin (pi/6))/(cos (pi/6))#
Equation (2) --># sin x.cos ((pi)/6) - sin ((pi)/6).cos x = sin (x - pi/6) = 0#
#sin (x - pi/6) = 0# --> #x = 0; x = pi; x = 2pi#
a. #x - pi/6 = 0# --> #x = pi/6#
b.# x - pi/6 = pi# --> #x = pi + pi/6 = (7pi)/6#
c. #x - pi/6 = 2pi# --> #x = 2pi + pi/6 = (pi)/6#
Check by calculator
#x = pi/6# -> #sqrt3sin x = sqrt3/2#; #cos (pi/6 - pi/3) = cos (-pi/6) = sqrt3/2#. OK
