Assuming the curve is
y = cos(x) - √3 sin(x)
differentiate once to get dy/dx, which gives the slope of the tangent line to y at some point x. The derivative is
dy/dx = -sin(x) - √3 cos(x)
The tangent line is horizontal when the derivative is 0, so you end up having to solve
-sin(x) - √3 cos(x) = 0
sin(x) = -√3 cos(x)
sin(x)/cos(x) = tan(x) = -√3
x = arctan(-√3) + nπ
(where n is an integer)
x = -arctan(√3) + nπ
x = -π/3 + nπ
You get two solutions in the interval [0, 2π] when n = 1 and n = 2, for which
x = 2π/3 or x = 5π/3