your equation is csc(x) = 1
since csc(x) is equal to 1/sin(x), then you get 1/sin(x) = 1
solve for sin(x) to get sin(x) = 1
when sin(x) = 1, x = 90 degrees.
since sine(x) is positive in the first and second quadrants only, then csc(x) is also positive in the first and second quadrants only.
csc(x) = 1 occurs when x = 90 and 90 is in the first and second quadrants, or right between them, i.e. it straddles the fence between first and second quadrant, therefore occurs only once in the first and second quadrants because it is at the intersection of both.
this all depends on how you define first and second quadrant.
if first is 0 to 90 and second is 90 to 180, then 90 is in both the first and second quadrant.
therefore csc(x) = 1 occurs only once every 360 degrees.
your formula therefore becomes:
csc(x) = 1 for x = 90 plus or minus k * 360 degrees where k is a positive integer.
here's a graph that shows some of the occurrences of the intersection of y = csc(x) and y = 1.
you can see that they go on indefinitely in both a positive and negative direction.
csc(x) = 1 when x = 90 degrees.
csc(x) = 1 when x = -4950 degrees which is equal to 90 minus 14 * 360 when k = 14.
csc(x) = 1 when x = 5850 degrees which is equal to 90 plus 16 * 360 when k = 16.
you can go on indefinitely in both direction since there is no restriction in the interval where csc(x) = 1 can occur.
here'a closeup graph that shows you that csc(x) = 1 only occurs once within each 360 degree cycle.
the vertical black lines are the demarcation between each 360 degree interval.