By definition of tangent,
tan(2θ) = sin(2θ) / cos(2θ)
Recall the double angle identities:
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos²(θ) - sin²(θ) = 2 cos²(θ) - 1
where the latter equality follows from the Pythagorean identity, cos²(θ) + sin²(θ) = 1. From this identity we can solve for the unknown value of sin(θ):
sin(θ) = ± √(1 - cos²(θ))
and the sign of sin(θ) is determined by the quadrant in which the angle terminates.
We're given that θ belongs to the third quadrant, for which both sin(θ) and cos(θ) are negative. So if cos(θ) = -4/5, we get
sin(θ) = - √(1 - (-4/5)²) = -3/5
Then
tan(2θ) = sin(2θ) / cos(2θ)
tan(2θ) = (2 sin(θ) cos(θ)) / (2 cos²(θ) - 1)
tan(2θ) = (2 (-3/5) (-4/5)) / (2 (-4/5)² - 1)
tan(2θ) = 24/7