Question 1007411
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{{{csc^2(2x)}}} - {{{csc(2x)}}} = {{{2}}}.

Introduce new variable y = csc(2x).

Then your equation takes the form 

{{{y^2 - y -2}}} = {{{0}}}.

Factor it:

(y+1)*(y-2) = 0

The solutions are y = -1 and y = 2.

Now you need to solve two equations:

1) csc(2x) = -1 -----> {{{1/sin(2x)}}} = -1 -----> sin(2x) = -1 -----> 2x = 270° -----> x = 135°.

2) csc(2x) = 2 -----> {{{1/sin(2x)}}} = 2 -----> sin(2x) = {{{1/2}}} -----> 2x = 30° and 2x = 150° -----> x = 15° and x = 75°.

<U>Answer</U>. The solutions are x = 15°, x = 75° and x = 135°.
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