To solve the problem, we need to use several trigonometric identities and principles.
1. From the given problem, we know that cosx = 8/17 and sinx < 0. To find the value of sinx, we use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
By rearranging the formula we get sin(x) = ±sqrt(1 - cos^2(x)).
2. Since sinx < 0 and cosx > 0 (cosx = 8/17), we're in the 4th quadrant, where sine values are negative. So, we take the negative square root.
3. Now plugging the value of cosx into our equation, we got sin(x) = -√(1 - (8/17)^2) = -0.8823529411764706
4. To find tanx, note that tanx = sinx / cosx. So tanx = -0.8823529411764706/ 8/17 = -1.875
5. Finally, we need to find tan(2x). We can use the double-angle identity for this: tan(2x) = 2tan(x) / (1 - tan^2(x))
6. Now, plug the value of tanx to find tan(2x) = 2*(-1.875)/(1 - (-1.875)^2) = 1.4906832298136645
Therefore, tanx = -1.875, sinx = -0.8823529411764706
Therefore, tan(2x) = 1.4906832298136645.