Question

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

Answer 1

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.