(2 tan x ) / (3 - 2 tan squared x) = 1?

Question

For interval 0, 2 pie

Answer 1

(2 tan x ) / (3 - 2 tan squared x) = 1

Multiply both sides by (3 - 2 tan squared x) and get

2 tan x = 3 - 2 tan squared x

Let u= tan x and the equation becomes

2 u= 3 - 2 u^2
2u^2 +2u -3 = 0
u^2 + u -1.5=0

completing the square you get
(u + 0.5)^2 -1.5 - 0.25 = 0
(u + 0.5)^2 -1.75= 0
(u + 0.5)^2 =1.75
u+0.5 = square root (1.75)
u +0.5=1.32287566
u=1.32287566-0.5
u=0.82287566

tan x = 0.82287566

x=arctan ( 0.82287566)
x=0.688534708 radians

as the tan shape repeats

x = 0.688534708 radians AND (0.688534708 +pi) radians
x = 0.688534708 and 3.83012736 radians

Answer 2

(2tan x)/(3 - 2 tan^2 x) = 1
2 tanx = 3 - 2 tan^2 x
2 tan^2x - 2tan x - 3 =0
tan^2 x -tan x -1.5 =0
Now let y= tan x
Then y^2 -y -1.5=0
y= [1+sqrt(1+6)]/2 = 0.5 +0.5sqrt 7
and y also = 0.5 -0.5sqrt 7
So tan x = 0.5 + 0.5sqrt 7 and
tan x = 0.5 -0.5 sqrt 7
So you will get 4 angles two positive tan x in quadrants I and III and two negative in quadrants II and IV.

You can use the arctan function on your caculator to approximate the answers.