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

2 Answers
Mar 21, 2018

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^@#

Mar 21, 2018

#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"#

http://www.nabla.hr/FU-TrigonometFunct3.htm