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- 1 month ago
Recall that csc(x)^2 - cot(x)^2 = 1

2 * csc(x)^2 - 5 * cot(x) = -1

2 * (cot(x)^2 + 1) - 5 * cot(x) = -1

2 * cot(x)^2 - 5 * cot(x) + 2 + 1 = 0

2 * cot(x)^2 - 5 * cot(x) + 3 = 0

Now use the quadratic formula to solve for cot(x)

cot(x) = (5 +/- sqrt(25 - 24)) / 4

cot(x) = (5 +/- 1) / 4

cot(x) = 6/4 , 4/4

cot(x) = 3/2 , 1

tan(x) = 2/3 , 1

x = arctan(2/3) + pi * k , pi/4 + pi * k

k is an integer

pi/4 , 5pi/4 , arctan(2/3) , arctan(2/3) + pi

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