text: Is the equation cos x - sqrt[ 3 sin x] = 1?\nOr is it cos x - sqrt[3] sin x = 1?\n\nI will treat each question separately.\n\n1. Observe that the cosine function cannot return values higher than 1, and the square root function cannot return negative values. Therefore, we can conclude that the equation is only true under the following conditions:\n\ncos x = 1\nsqrt[3 sin x] = 0\n\ncos x = 1 and sin x = 0\n\nThe values of x satisfying these two equations are 0 and 2 pi.\n\n2.\n\ncos x - sqrt[3] sin x = 1\ncos x = 1 + sqrt[3] sin x\ncos^2 x = 1 + 2 sqrt[3] sin x + 3 sin^2 x\n1 - sin^2 x = 1 + 2 sqrt[3] sin x + 3 sin^2 x\n4 sin^2 x + 2sqrt[3] sin x = 0\n\n2 sin x (2 sin x - sqrt[3] ) = 0\n\nsin x = 0 or sqrt[3] / 2\n\nFor the first equation, the possible values of x are 0, pi or 2 pi.\n\nFor the second equation, look at the 30-60-90 right triangle to determine that x is either pi / 3 or 2 pi / 3 (60 or 120 degrees).\n\nSo, all of the possible values for x are 0, pi / 3, 2 pi / 3, pi, and 2 pi.