Question

How do you solve `csc^2x-cscx-2=0` between the interval `0<=x<=2pi`?

Answer 1

`csc^2 x - 2csc - 2 = 0`
Solve this quadratic equation for csc x.
Since (a - b + c = 0), use shortcut. The 2 real roots are csc x = -1 and `csc x = -c/a = 2`
Convert `csc = 1/(sin)` to sin x, we get sin x = - 1 and `sin x = 1/2`
a. sin x = - 1 --> x = (3pi)/2
b.`sin x = 1/2`.
Trig table and unit circle give:
`x = pi/6` and `x = (5pi)/6`
Answers for `(0, 2pi)`
`pi/6; (3pi)/2, (5pi)/6`