Solve for x using trigonometric identities?

Question

(cot x)+ √(3) = (csc x)

solve for x over [0, 2π).

the answer is x = (2π)/(3)
two pi over three

how do i get to the answer?



the problem spelled out:
cotangent of x, plus the square root of three, equals cosecant of x.
solve for x over zero(included) to two pi (excluded).

thank you if you can help..

Answer 1

cot x + √3 = csc x
csc x - cot x = √3
1/sinx - cosx/sinx = √3
(1-cosx)/sinx = √3
sq both side

(1-cosx)^2/sin^2x= 3
1 - 2cosx + cos^2x = 3 sin^2 x
1 - 2cosx + cos^2x = 3(1-cos^2x)
rearrange the eq
4cos^2x - 2cosx - 2 = 0
(4cosx + 2)(cosx - 1) = 0
cosx = -1/2 or 1

sub x = 1 to the original eq does not give a correct answer
so x = 120 or 2pi/3