Evaluate the values of x for tan^2x=(1+tanx)/2

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We'll re-write the expression:

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

We'll subtract tan x both sides:

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**

We'll put the next factor as 1:

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}**

We have to determine x given that (tan x)^2 = (1 + tan x) / 2.

Now (tan x)^2 = (1 + tan x) / 2

multiply both sides by 2

=> 2 (tan x)^2 = 1 + tan x

=> 2 (tan x)^2 - 1 - tan x =0

=> 2 (tan x)^2 - 2 tan x + tan x -1 =0

=> 2 tan x ( tan x -1 ) + 1( tan x -1) =0

=> (2 tan x + 1)( tan x - 1) =0

=> (2 tan x + 1) = 0 or ( tan x - 1) =0

=> tan x = -1/2 or tan x = 1

Therefore x can take the values arc tan -1/2 and arc tan 1

or x = -26.56 + n*180 degrees or 45 + n*180 degrees.

**The required values are x = -26.56 + n*180 degrees or 45 + n*180 degrees.**

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