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Use The Double-Angle Identities To Find Tan(2x) If Cosx=(8)/(17) And Sinx<0.

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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.

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