Solve for 1+tan2 x=2tanx for 0<=x<=360

Start with the given equation:
1 + tan^2(x) = 2tan(x)
Move all terms to one side to form a quadratic equation:
1 + tan^2(x) - 2tan(x) = 0
Rearrange in standard quadratic form ay^2 + by + c = 0, where y = tan(x):
tan^2(x) - 2tan(x) + 1 = 0

This is a perfect square trinomial, which can be factored as (a - b)^2 = a^2 - 2ab + b^2.
Here, a = tan(x) and b = 1:
(tan(x) - 1)^2 = 0

Take the square root of both sides:
tan(x) - 1 = 0
Add 1 to both sides:
tan(x) = 1

The tangent function is positive in the first and third quadrants.

• First Quadrant: The angle whose tangent is 1 is 45°. x = 45°
• Third Quadrant: The angle in the third quadrant is 180° plus the reference angle. x = 180° + 45° = 225°

Therefore, the solutions for x in the range 0° <= x <= 360° are 45° and 225°.