From the figure, we immediately have
cos(θ) = 8/17
sin(θ) = 15/17
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
Then
tan(2θ) = (2 sin(θ) cos(θ)) / (2 cos²(θ) - 1)
tan(2θ) = (2 × 15/17 × 8/17) / (2 × (8/17)² - 1)
tan(2θ) = -240/161
