1 + tanx / 1 + cotx = 2

x = tan^ (-1) ((i sqrt (3)) / 2 + 1/2) + π n_1 for n_1 element Z

or x = tan^ (-1) ( - (i sqrt (3)) / 2 + 1/2) + π n_2 for n_2 element Z

Step-by-step explanation:

Solve for x:

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

Multiply both sides of 1 + cot (x) + tan (x) = 2 by tan (x):

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

Subtract 2 tan (x) from both sides:

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

Subtract 1 from both sides:

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

Add 1/4 to both sides:

1/4 - tan (x) + tan^2 (x) = - 3/4

Write the left hand side as a square:

(tan (x) - 1/2) ^2 = - 3/4

Take the square root of both sides:

tan (x) - 1/2 = (i sqrt (3)) / 2 or tan (x) - 1/2 = - (i sqrt (3)) / 2

Add 1/2 to both sides:

tan (x) = 1/2 + (i sqrt (3)) / 2 or tan (x) - 1/2 = - (i sqrt (3)) / 2

Take the inverse tangent of both sides:

x = tan^ (-1) ((i sqrt (3)) / 2 + 1/2) + π n_1 for n_1 element Z

or tan (x) - 1/2 = - (i sqrt (3)) / 2

Add 1/2 to both sides:

x = tan^ (-1) ((i sqrt (3)) / 2 + 1/2) + π n_1 for n_1 element Z

or tan (x) = 1/2 - (i sqrt (3)) / 2

Take the inverse tangent of both sides:

Answer: x = tan^ (-1) ((i sqrt (3)) / 2 + 1/2) + π n_1 for n_1 element Z

or x = tan^ (-1) ( - (i sqrt (3)) / 2 + 1/2) + π n_2 for n_2 element Z