Determining the solutions of: `3*csc^2(x-pi/3)-5*csc(x-pi/3) = 2` ?

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The equation `3*csc^2(x-pi/3)-5*csc(x-pi/3) = 2` has to be solved for x.

`3*csc^2(x-pi/3)-5*csc(x-pi/3) = 2`

Let y = `csc (x - pi/3)`

=> `3*y^2 - 5y = 2`

=> `3y^2 - 6y + y - 2 = 0`

=> `3y(y - 2) + 1(y - 2) = 0`

=> `(3y + 1)(y - 2) = 0`

=> y = `-1/3` and y = 2

As y =` csc (x - pi/3)`

=> `sin (x - pi/3) = -3` and `sin(x - pi/3) = 1/2`

As sin x lies in [-1, 1], the root -3 can be ignored.

=> `sin (x - pi/3) = 1/2`

=> `sin (x - pi/3) = sin (pi/6 + n*2*pi)`

=> `x - pi/3 = pi/6 + n*2*pi`

=> `x = pi/2 + n*2*pi`

The solution for x is `pi/2 + n*2*pi`

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