We have
tan(2x) = sin(2x) / cos(2x)
tan(2x) = (2 sin(x) cos(x)) / (2 cos²(x) - 1)
Recall that
sin²(x) + cos²(x) = 1
and since sin(x) is negative, it follows that
sin(x) = - √(1 - (2/3)²) = -√5/3
Then
tan(2x) = (2 (-√5/3) (2/3)) / (2 (2/3)² - 1) = 4√5
