For what value of x is the following 2(tan x)^2+tan(-x)=1 true

For the beginning, we'll notice that one terms has the
opposite variable, -x. Since the tangent function is odd, we'll write the
term:

tan(-x) = - tan x

We'll
re-write the given expression:

2 (tan x)^2 - tan x =
1

We'll factorize by tan
x:

tan x(2 tan x - 1) =
1

We'll put tan x =
1

x = pi/4 +
k*pi

2 tan x - 1 =
1

We'll add 1 both sides:

2
tan x = 2

We'll divide by
2:

tan x = 1

x =
pi/4 +
k*pi

or

tan x =
-1

The tangent fucntion is negative when x is in the second
or the fourth quadrant.

x = pi -
pi/4

x = 3pi/4 +
k*pi

x = 2pi -
pi/4

x = 7pi/4 +
k*pi

The solutions of the
equation are:

{pi/4 + k*pi ;
3pi/4 + k*pi ; 7pi/4 + k*pi}