Question

How do you find the general solutions for `sec^2x-secx-2=0`?

Answer 1

Solve `sec^2 x - sec x - 2 = 0`

Ans: `pi, +- pi/3`

Explanation:

Call sec = t, we get a quadratic equation:

`t^2 - t - 2 = 0.`
Since (a - b + c = 0), use the shortcut. The 2 real roots are: t = - 1 and
`t = - c/a = 2.`

a. `t = sec x = 1/cos x = - 1` --> `cos x = - 1`--> `x = pi`

b. `t = sec x = 1/cos x = 2` --> `cos x = 1/2` --> `x = +- pi/3`

General solutions:
`x = pi + 2kpi = (2k+ 1)pi`
`x = +- pi/4 + 2kpi`