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Given that sin ×=5/13, 0°<x<90°,evaluate cos×-sin×\2tan ×​

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Here is the correct solution:

Given that sin x = 5/13, we can use the Pythagorean identity cos²x + sin²x = 1 to find cos x as follows:

cos²x + (5/13)² = 1
cos²x = 1 - (5/13)²
cos x = ±sqrt(1 - (5/13)²)

Since 0° < x < 90°, we know that cos x > 0. Therefore, we can take the positive square root:

cos x = sqrt(1 - (5/13)²)
cos x = 12/13

Next, we can evaluate sin x / (2tan x) as follows:

sin x / (2tan x) = (5/13) / (2sin x / cos x)
sin x / (2tan x) = (5/13) * (cos x / 2sin x)
sin x / (2tan x) = (5/13) * (cos x / (2 * (5/13)))
sin x / (2tan x) = cos x / 2

Substituting the value of cos x that we found earlier, we get:

sin x / (2tan x) = (12/13) / 2
sin x / (2tan x) = 6/13

Finally, we can evaluate cos x - sin x / (2tan x) as follows:

cos x - sin x / (2tan x) = (12/13) - (5/13) / (2 * (5/13))
cos x - sin x / (2tan x) = 12/13 - 1/13
cos x - sin x / (2tan x) = 11/13

Therefore, cos x - sin x / (2tan x) = 11/13.
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