The given function is \( \displaystyle y = \sqrt{\frac{1+\sin x}{1-\sin x}} \) and we need to find the derivative of this function with respect to \( x \).
Solution:
Step 1: Simplify the Expression
\( \displaystyle y = \sqrt{\frac{1 + \sin x}{1 – \sin x}} \\ \displaystyle = \sqrt{\frac{(1 + \sin x)^2}{(1 – \sin x)(1 + \sin x)}} \\ \displaystyle = \sqrt{\frac{(1 + \sin x)^2}{1 – \sin^2 x}} \\ \displaystyle = \sqrt{\frac{(1 + \sin x)^2}{\cos^2 x}} \\ \displaystyle = \sqrt{\frac{(1 + \sin x)^2}{\cos^2 x}} \\ \displaystyle = \frac{1 + \sin x}{\cos x} = \sec x + \tan x \\ \)Step 2: Differentiate the Simplified Form
\( \displaystyle \frac{dy}{dx} = \frac{d}{dx} (\sec x) + \frac{d}{dx} (\tan x) \\ \displaystyle = \sec x \tan x + \sec^2 x = \sec x (\tan x + \sec x) \\ \)Thus, \(\displaystyle \boxed{\frac{dy}{dx} = \sec x (\tan x + \sec x)} \)
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