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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