tan(x)^2 - 4*tan(x) + 1 = 0 let tan (x) = a
a^2 - 4a + 1 = 0 complete the square on a
a^2 - 4a + 4 = - 1 + 4
(a - 2)^2 = 3 take both roots
(a - 2) = ±√3
a = 2 + √3 a = 2 - √3
So either
tan (x) = 2 + √3
arctan ( 2 + √3) = x = 75°
So sin (2 * 75°) = sin (150°) = 1/2
Or
tan (x) = 2 - √3
arctan ( 2 - √3) = x = 15°
So sin (2 * 15°) = sin (30°) = 1/2