tan x = 2 sin x I am going to use x since it is quicker for me to type.
tan x - 2 sin x = 0 move everything over to one side
(sin x)/(cos x) - 2 sin x = 0 write tangent in terms of sines and cosines
sin x (1/cos x - 2) = 0 factor out sine
sin x = 0 or 1/cos x - 2 = 0 set each factor equal to zero
sin x = 0 what angles give you a sine value of zero? x = n * pi where n is an integer.
We are looking for angles in [0, 2pi) so 0*pi and 1*pi are the only angles in that interval. So, 0 and pi are two of our solutions.
1/cos x - 2 = 0 cos x = 1/2 your reference angle is pi/3
cosine is positive in quadrants 1 and 4, so pi/3 and 5*pi/3 are the angles in our interval.
x = 0, pi, pi/3 and 5*pi/3 are the solutions.