# `sec^2 (x) + tan(x) - 3 = 0` Use inverse functions where needed to find all solutions of the equation in the interval `0,2pi)`.

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### 2 Answers

`sec^2(x)+tan(x)-3=0`

use the identity `sec^2(x)=1+tan^2(x)`

`1+tan^2(x)+tan(x)-3=0`

`tan^2(x)+tan(x)-2=0`

solve using quadratic formula,

`tan(x)=(-1+-sqrt(1^2-4*1*(-2)))/2`

`tan(x)=(-1+-sqrt(9))/2=(-1+-3)/2=1,-2`

General solutions for tan(x) =1 are x=pi/4+pin

Solutions for the `0<=x<=2pi` range are,

x=pi/4 , x=(5pi)/4

Solutions for tan(x)=-2 for the range are,

`x=pi-arctan(2) , 2pi-arctan(2)`

Solutions are

x= `pi/4` , `(5pi)/4`, 2.034 , 5.176 radians

`sec^2x + tanx - 3 = 0 `

Change `sec^2x` to `tan^2x+1`

`tan^2x + 1 + tanx - 3 = 0 `

Group like terms

`tan^2x + tanx - 2 = 0 `

Factor

`(tanx + 2)(tanx - 1) = 0 `

Set each part equal to 0

`tanx = -2 or tanx = 1 `

Use arctan to solve for x

`x = 2.034` radians or `x = 5.176` radians or `x = pi/4` radians or x = `5pi/4` radians