Simplify: \( \frac{(\sec x-\tan x)(\sec x+\tan x)}{\tan x} \)

To simplify the expression \( \frac{(\sec x-\tan x)(\sec x+\tan x)}{\tan x} \), we start by recognizing that the numerator is a difference of squares: \[ (\sec x - \tan x)(\sec x + \tan x) = \sec^2 x - \tan^2 x \] Now, we can apply the Pythagorean identity, which states that \( \sec^2 x - \tan^2 x = 1 \): \[ \sec^2 x - \tan^2 x = 1 \] This means that our expression can now be simplified to: \[ \frac{1}{\tan x} \] Recalling the definition of cotangent, we have \( \cot x = \frac{1}{\tan x} \). Therefore, we can rewrite the expression as: \[ \cot x \] Thus, the simplified form of the original expression is: \[ \boxed{\cot x} \]