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

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Answer 1 · 2018-03-21T03:02:43

see below.

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 · 2018-03-21T03:09:03

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

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

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 $tan x = +-1$

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

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

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

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