Question

How do you find all solutions of the equations `sec^2x-secx=2` in the interval `[0,2pi)`?

Answer 1

There are three solutions to this equation:

`x_1 = \pi/3; \qquad x_2 = (5\pi)/3;` and `x_3 = \pi;`

Explanation:

Setting `t=\secx`, we get the following quadratic equation:

`t^2-t-2=0`

which has as solutions, `t_1 = 2; \qquad t_2 = -1;`

So we need to look for solutions to the equations -
[1] `\secx = 2 : \qquad \cosx = 1/2 \rightarrow x_1 = \pi/3; \quad x_2 = (5\pi)/3;`

[2] `\secx = -1: \qquad \cosx=-1 \rightarrow x_3 = \pi;`