Final answer:
tan(x) = 10/11 in the first quadrant allows us to find cos(2x) using the identity cos(2x) = 1 - 2sin²(x). After calculating the sine of x, we find that cos(2x) = -21/221, which is not one of the options provided.
Step-by-step explanation:
If we have tan(x) = 10/11 and x is in the first quadrant, we need to find cos(2x). Since tan(x) = opposite/adjacent, this allows us to form a right-angled triangle where the opposite side to angle x is 10 units long and the adjacent side is 11 units long. Using the Pythagorean theorem, the hypotenuse (h) would then be √(10² + 11²) = √(100 + 121) = √221.
Now, using the identity cos(2x) = 1 - 2sin²(x), we can find sin(x) = opposite/hypotenuse = 10/√221. Then sin²(x) = 100/221. Plugging this into the identity gives us cos(2x) = 1 - 2(100/221) = (221/221) - (200/221) = 21/221. However, since cos(2x) will actually be negative in the second quadrant (as 2x will be greater than 90° but less than 180°), the correct answer is -21/221, which is not one of the given options.