To solve this equation, we set each factor equal to zero and solve for x:

Setting tg x + 1 = 0:
tg x = -1
x = arctan(-1) + kπ, where k is an integer

Setting 2 cos x/3 - sqrt3 = 0:
cos x = 3sqrt3/2
x = arccos(3sqrt3/2) + 2πk or x = -arccos(3sqrt3/2) + 2πk, where k is an integer

Therefore, the solutions to the equation are:
x = arctan(-1) + kπ, where k is an integer
x = arccos(3sqrt3/2) + 2πk or x = -arccos(3sqrt3/2) + 2πk, where k is an integer