Question

How do you solve `sec^2x-2tan^2x=0` in the interval [0,360]?

Answer 1

see below.

Explanation:

First, using Phytagorean identities trigonometric,
`sec^2 x=tan^2 +1`

Then,
`(tan^2 x+1)-2 tan^2 x=0`
`-tan^2 x+1=0`
`tan^2 x=1`
`x=tan^(-1) (sqrt(1))`
`x=45^@ and x=225^@`

Answer 2

`x = color(blue)(45^@, 225^@ " for " pi/4), color(green)(135^@, 315^@ " for " -pi/4`

`"in the interval [0,360] as tan is positive in I & III quadrants"`

Explanation:

To solve `sec^2 x - 2 tan^2x = 0, "in the interval [0,360]"`

`sec*2x = 1 + tan^2x, "Identity"`

`:. 1 + canceltan^2x -cancel( 2 tan^2x)^color(red)(tan^2x) = 0`

`1 - tan^2x = 0`

`tan^2x = 1` or `tan x = +-1`

`x = +- (pi/4)^c`

`x = color(blue)(45^@, 225^@ " for " pi/4), color(green)(135^@, 315^@ " for " -pi/4`

`"in the interval [0,360] as tan is positive in I & III quadrants only"`