Final answer:
Using the given cosx=-2/5 with x in the fourth quadrant, we find the opposite side of the triangle, calculate tan(x), and then use the double angle formula to find that the exact value of tan(2x) is 4/3.
Step-by-step explanation:
If cosx = -2/5 with x being an angle whose terminal side is in the fourth quadrant, we can determine the exact value of tan(2x). In the fourth quadrant, cosine is negative and sine is also negative.
To find tan(2x), we use the double angle formula for tangent which is tan(2x) = 2tan(x) / (1 - tan^2(x)). However, we first need to find tan(x) using the given cosine value and the Pythagorean theorem.
Since cos(x) = adjacent/hypotenuse, we have the adjacent side as -2 and the hypotenuse as 5. We can find the opposite side using (opp)^2 + (-2)^2 = 5^2, which gives us the opposite side is 1 or -1. Because we are in the fourth quadrant, the opposite side should be negative, so opp = -1. We now have all sides of the triangle.
Next, we find tan(x) = opposite/adjacent = -1/(-2) = 1/2. Then, we apply the double angle formula: tan(2x) = 2(1/2) / (1 - (1/2)^2) = 1 / (1 - 1/4) = 1 / (3/4) = 4/3.
The exact value of tan(2x) when cosx = -2/5 and x is in the fourth quadrant is 4/3.