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Recall the Pythagorean identity,
cos²(x) + sin²(x) = 1
and the definition of tangent,
tan(x) = sin(x) / cos(x)
Given that tan(α) < 0 and cos(α) = -5/13 < 0, it follows that sin(α) > 0. Then using the identity above, we find
sin(α) = √(1 - cos²(α)) = 12/13
Now recall the double angle identity for sine,
sin(2x) = 2 sin(x) cos(x)
Then
sin(2α) = 2 sin(α) cos(α) = 2 • 12/13 • (-5/13) = -360/169