(1)/(sec x-tan x)-(1)/(sec x+tan x)

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Given Expression: We are given the expression $\frac{1}{\sec x - \tan x} - \frac{1}{\sec x + \tan x}$. To simplify this expression, we will find a common denominator and combine the fractions.Find Common Denominator: The common denominator for the two fractions $\frac{1}{\sec x - \tan x}$ and $\frac{1}{\sec x + \tan x}$ is $(\sec x - \tan x)(\sec x + \tan x)$. We will multiply the numerator and denominator of each fraction by the conjugate of its denominator to get the common denominator.Multiply by Conjugate: For the first fraction, we multiply the numerator and denominator by $\sec x + \tan x$: $\frac{1}{\sec x - \tan x} \cdot \frac{\sec x + \tan x}{\sec x + \tan x} = \frac{\sec x + \tan x}{(\sec x - \tan x)(\sec x + \tan x)}$.Combine Fractions: For the second fraction, we multiply the numerator and denominator by $\sec x - \tan x$: $\frac{1}{\sec x + \tan x} \cdot \frac{\sec x - \tan x}{\sec x - \tan x} = \frac{\sec x - \tan x}{(\sec x + \tan x)(\sec x - \tan x)}$.Subtract Numerators: Now we combine the two fractions with the common denominator: $\frac{\sec x + \tan x}{(\sec x - \tan x)(\sec x + \tan x)} - \frac{\sec x - \tan x}{(\sec x + \tan x)(\sec x - \tan x)}$.Simplify Numerator: We subtract the numerators and keep the common denominator: $\frac{(\sec x + \tan x) - (\sec x - \tan x)}{(\sec x - \tan x)(\sec x + \tan x)}$.Denominator Simplification: Simplify the numerator by combining like terms: $\sec x + \tan x - \sec x + \tan x = 2\tan x$.Apply Pythagorean Identity: The denominator is the product of a binomial and its conjugate, which is a difference of squares: $(\sec x - \tan x)(\sec x + \tan x) = \sec^2 x - \tan^2 x$.Final Simplified Expression: We know that $\sec^2 x - \tan^2 x = 1$ from the Pythagorean identity for secant and tangent: $\sec^2 x = 1 + \tan^2 x$, so $\sec^2 x - \tan^2 x = 1$.Now we can write the simplified expression: $\frac{2\tan x}{1} = 2\tan x$.

$\left(\frac{1}{\sec x - \tan x}\right) - \left(\frac{1}{\sec x + \tan x}\right)$

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(1sec⁡x−tan⁡x)−(1sec⁡x+tan⁡x)\left(\frac{1}{\sec x - \tan x}\right) - \left(\frac{1}{\sec x + \tan x}\right)(secx−tanx1​)−(secx+tanx1​)

$\left(\frac{1}{\sec x - \tan x}\right) - \left(\frac{1}{\sec x + \tan x}\right)$