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
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
