How do you solve cot^2 x +csc x = 1 from [0,360]?

1 Answer
Nghi N.
Jul 22, 2015

Solve: cot^2 x + csc x = 1

Ans: pi/2; (7pi)/6; and (11pi)/6

Explanation:

cos^2 x/sin^2 x + 1/sin x = 1

cos^2 x + sin x = sin^2 x
(1 - sin^2 x) + sin x = sin^2 x

2sin^2 x - sin x - 1 = 0
Case (a + b + c = 0), the 2 real roots are: sin x = 1 and sin x = -1/2

a. sin x = 1 --> x = pi/2

b. sin x = - 1/2 --> x = (7pi)/6 and x = (11pi)/6

Within interval (0, 2pi), 3 answers: pi/2; (7pi)/6 and (11pi)/6.

Check with x = (7pi)/6.
cot (7pi)/6 = sqrt3 --> cot^2 ((7pi)/6) = 3.
csc ((7pi)/6) = 1/sin ((7pi)/6) = - 2
cot ((7pi)/6) - csc ((7pi)/6) = 3 - 2 = 1 Correct.

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