2tan x = 1 - tan²x
=> Add tan²x to both sides, then
2tan x + tan²x = 1
=> Let tan x be p, then
2p + p² = 1
p² + 2p = 1
=> Using completing the square method, half the co-efficient of p and then square it, i.e.
2 → 2 * 1/2 = 1 → (1)² = 1
=> Add 1 to both sides in p² + 2p = 1, i.e.
p² + 2p + 1 = 1 + 1
(p + 1)² = 2
=> Square root both sides, then
p + 1 = ± √2
p = - 1 + √2 or p = - 1 - √2
=> Remember that tan x is p, then
tan x = - 1 + √2 or tan x = - 1 - √2
tan x = 0.4142 or tan x = - 2.4142
=> Starting with tan x = 0.4142, shift tan from the left to the right and it automatically becomes arc-tan, i.e.
x = tan^-1 (0.4142)
x = 22.5° or π/8
=> In the circle of 4 quadrants, the value of the tan measure either in degree or radian is +ve at the 3rd quadrant, then to obtain this value you make use of the formula:
3rd quadrant: 180° + x, where x is 22.5°, then 180° + 22.5° = 202.5° or 9π/8.
=> Moving on to tan x = - 2.4142, you'll have
x = tan^-1 (- 2.4142)
x = - tan^-1 (2.4142)
=> Here, the arc-tan of 2.4142 is 67.5°, then let x be 67.5° but arc-tan is -ve denoting that the value to be determined must come from the 2nd & 4th quadrants of the unit circle, where the tan measure of the angles are -ve, then to accomplish this, you make use of the formulae:
* 2nd quadrant: 180° - x, where x is 67.5°, then 180° - 67.5° = 112.5° or 5π/8.
* 4th quadrant: 360° - x, where x is 67.5°, then 360° - 67.5° = 292.5° or 13π/8.
=> Since we aren't given any range of values for which other values of x can be obtained, we state that x = 22.5°, 112.5°, 202.5° or 292.5° in degree measure and also, x = π/8, 5π/8, 9π/8 or 13π/8 in radian measure.